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Two-compartment stochastic model of a neuron
Physica D 132 (1999) 267–286
Two-compartment stochastic model of a neuron Petr Lánský a,∗ , Roger Rodriguez b a
b
Institute of Physiology, Academy of Sciences of the Czech Republic, Videñská 1083, 142 20 Prague 4, Czech Republic Centre de Physique Théorique, CNRS and Faculté des Sciences de Luminy, Université de la Méditerranée, Luminy-Case 907, F-13288 Marseille Cedex 09, France Received 11 August 1998; received in revised form 7 February 1999; accepted 12 February 1999 Communicated by A.M. Albano
Abstract A two-compartment neuronal model is investigated. The model neuron is composed of two interconnected parts – a dendrite and a trigger zone, with a white noise input in the dendritic compartment. The first and the second moments of the stochastic processes describing the membrane depolarization in both compartments are derived and investigated. When a firing threshold is not imposed, the level of neuronal activity is deduced from a generally accepted relationship between the membrane potential and the firing frequency. When a firing threshold is imposed, some approximations and simulations are used to characterize the spiking activity of the model. It is shown that the activity of the two-compartment model is less sensitive to abrupt changes in stimulation than the activity of the single-compartment model. The delayed response of the complex model is a natural consequence of the fact that the input takes place in the compartment different from that at which the output is generated. Further, the model predicts serial correlation of interspike intervals, which is a phenomenon often observed in experimental data but not reproducible in the single-compartment models under steady-state stimulation. Finally, the investigated model neuron shows lower sensitivity to the input intensity and larger coding range than the single-compartment model. ©1999 Elsevier Science B.V. All rights reserved. PACS: 87.10.+e; 05.40.+j Keywords: Neuron model; Integrate-and-fire; Stochastic diffusion process; Membrane depolarization; Bursting
1. Introduction There is an enormously wide range of approaches to the modeling of neuronal activity starting with deterministic biophysical concepts which have proved to be very successful in explaining the generation of the various types of membrane potentials, such as the Hodgkin–Huxley model of the action potential. On the other end of the spectrum, if considering the integrated activity of single neurons or assemblies of neurons, some drastic simplifications are necessary. The leaky integrate-and-fire, as the most simplified but still sufficiently realistic model of the Hodgkin–Huxley schema, appears to be the most suitable for this purpose [1,2]. However, both of these two very ∗
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0167-2789/99/$ – see front matter ©1999 Elsevier Science B.V. All rights reserved. PII: S 0 1 6 7 - 2 7 8 9 ( 9 9 ) 0 0 0 3 4 - 2
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different approaches have in common the fact that all the properties of the model neurons are concentrated into a single point in space (single-point models) and no geometrical architecture of the cell is considered. In contrast to this neuron representation are computerized multicompartment models (for a review see [3,4]) or cable models, either deterministic (for a review see [5]) or stochastic [6,7]. These spatially complex models usually do not aim at direct description of the input–output transfer but are mainly focused on the computational aspects of the information processing within the neuron itself. Besides simplifying the properties of the membrane along the way from Hodgkin–Huxley model to the leakyintegrator, and transforming the infinitely variable spatial structure into a single point, another set of assumptions deals with the properties of the neuronal input. An apparent variability in neuronal activity stresses the fact that if input–output transfer of a neuron or of a neuronal system is investigated, the stochastic phenomena may play an important role. The sources of randomness can be intrinsic as well as extrinsic. Nevertheless, it seems to be more likely that the stochastic variation of the input causes global changes and is crucial for the modification of the input–output information transfer characteristics [8]. The simplest approach to introduce external random influence into the neuronal models is considering a stochastic variant of the leaky integrate-and-fire model, which is based on two assumptions describing its behavior between two consecutive firings (resets). Firstly, the mean value of the membrane depolarization is described by the deterministic leaky integrate-and-fire model; secondly, the membrane potential at any given time instant is normally distributed around its mean value. The stochastic process fulfilling these requirements is an Ornstein–Uhlenbeck diffusion process. Stochastic models based on the leaky integrate-and-fire concept has been extensively used, modified and generalized (for a review see [9]). As stressed above, the tractability of these models follows from the fact that all the properties of the neuron are concentrated into a single point in space with none of the geometrical properties of a real neuron being taken into account, and it is a substantial simplification of reality. Therefore, in the recent years, several attempts appeared to generalize the single-point models making them more biologically relevant but still mathematically tractable. Kohn [10] proposed a two-compartment (two-point) model and similar models were further developed and studied [11,12]. The model we analyze here belongs to this class and it is based on the following set of hypotheses [13]: (i) The neuron is assumed to be made of two interconnected – dendritic and trigger zone – compartments. (ii) The input is present at the dendritic compartment only. (iii) The potentials of the two compartments are described by leaky integrators with a reset mechanism at the trigger zone. It follows from (i) that in this two-point model the dendritic potential depends on the trigger zone potential, a feature which was for the sake of tractability neglected in our previous papers [11,12]. The aim of this article is to investigate the first two statistical moments of the membrane depolarization in both compartments and to find their implications for the coding properties of the model. A comparison is made with activity evoked in the single-point model being stimulated by the same signal – constant or square pulse stimulation.
2. Single-point model Before going into the details about the two-point model, let us summarize the relevant information on its singlepoint counterpart (for a simple introduction to the single-point models see [14], a complete review can be found in [9]). One of the most common description of the stochastic evolution of the membrane potential under the singlepoint scenario is Stein’s model [15], which employs a Poissonian character of the timing of the incoming signals. It is an easy-to-interpret starting point from which more abstract Ornstein–Uhlenbeck model can be meaningfully derived (e.g. [16,17]). Despite that one could arrive at the stochastic leaky integrator just by adding the white noise
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to the input of the deterministic model, the approach via the Stein’s model is more illustrative and suitable for parameter interpretation (see Section 5). 2.1. Subthreshold behavior In the Ornstein–Uhlenbeck model, the behavior of the depolarization X of the membrane is described by the stochastic differential equation X (1) X(0) = X0 , dX = − + µ dt + σ dW, τ where W is a standard Wiener process, τ > 0 is the membrane time constant (τ = RC, where R is the membrane resistance and C is its capacitance), µ and σ > 0 are constants. The parameters µ and σ 2 in (1), expressed respectively in mV and mV2 per unit time, reflect the input signal and its variability resulting from the stochastic dendritic currents generated by sensory stimulation or by action of other neurons. The random initial depolarization defined by X0 was studied in [18] and recently in [19], but commonly it is assumed that Prob(X0 = 0) = 1. A direct generalization of model (1) assumes that parameters µ and σ 2 depend on time, reflecting time course of the signal impinging on the neuron (for discussion see [20]). Namely, constant σ and µ(t) = µ0 + µ1 cos(ωt), in addition to the reset after firing, is the most common generalization. This system produces stochastic resonance effects [21,22], a phenomenon which was expected and has been extensively searched for in the nervous system in the recent years (e.g. [23–25]). The mean and variance of X given by (1) are µ µ exp(−αt), (2) E(X(t)) = + E(X0 ) − α α σ2 σ2 + Var(X0 ) − exp(−2αt), (3) Var(X(t)) = 2α 2α where α = τ −1 . These quantities describe completely the behavior of the membrane depolarization because process (1) is Gaussian. Thus, from (2) and (3), at the steady-state, X(∞) ∼ N (µ/α, σ 2 /2α). From Eqs. (2) and (3) the relation between E(X(t)) and Var(X(t)) can be calculated. Denoting the asymptotic ratio between the absolute value of the mean depolarization and its variance by ρ1 , ρ1 = |E(X(∞))|/Var(X(∞)), we have ρ1 =
2|µ| σ2
(4)
and this value will be compared with an analogous one for the two-point model to show improvement in the signal-to-noise ratio achieved by introducing the second compartment. 2.2. Threshold Single-point model (1) is only appropriate for subthreshold response. If the neuron is capable of generating spike, a threshold condition has to be imposed. The firing of an action potential is identified with the first crossing of a firing threshold S, S > X0 and the time origin is the moment of the last firing. At the moments of spike generation the membrane potential is repeatedly reset to its initial value X0 . The reset of X after firing introduces a strong nonlinearity into the model. Further, it causes that no information about the input prior to firing prevails on the neuron. Therefore the interspike intervals are described by independent realizations of the random variable T = inf{t > 0; X(t) ≥ S|X0 < S}.
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Under the frequency coding scenario, which is plausible for constant and long-lasting stimulations, the shapes of the input–output frequency curves are used to characterize the models as well as the experimental data. In the latter case, the curve is obtained by plotting 1/t¯ versus the stimulus intensity, where t¯ is the average interspike interval. For the deterministic version of model (1), σ = 0, the length of the interspike interval can be easily calculated. It is (1/α) ln(µ/(µ − αS)) and the shape of the transfer function follows directly from it, ( −1 µ 1 ln , µ/α > S, + t ref α µ−αS (5) f (µ) = 0, µ/α ≤ S, where tref denotes an absolute refractory period preventing permanent increase of f with increasing µ, namely −1 in (5) µ → ∞ implies f → fmax = tref . Relating the maximum frequency to the absolute refractory period is only formal because in reality the saturation frequency limiting the activity of a neuron comes earlier for complex biophysical reasons. For σ > 0, the solution of the first-passage-time problem is not a simple task even for apparently simply looking model (1) and therefore numerical (e.g. [26,27]) and simulation [28] techniques have often been proposed. The shapes of the input–output frequency curves for model (1) can be found in [11], where the simulated first-passage times were treated in the same way as experimentally measured interspike intervals, which means that E(T ) was estimated by averaging over many realizations of the times when the simulated trajectories of X(t) crossed the firing threshold for the first time. Similarly, applying the numerical techniques, the curves are constructed by plotting 1/(E(T ) + tref ), where the mean first-passage time is calculated from available theoretical formulas [29,26]. Defining the coding range of a neuron, and similarly of a model neuron, as a set of µ which are characterized by different firing frequencies [30], the coding range of the deterministic counterpart of model (1) is enlarged by adding the noise. The firing activity of the noisy model starts at the levels below µ/α which is required by condition (5) for the deterministic model. In general, increasing the amplitude of the noise (σ in (1)), the transfer function gets broader and thus, in its left-hand part, becomes closer to the sigmoid function. 2.3. Sigmoid transfer function The above introduced transfer function (5) may be related to another one which is more common in neural network applications than in biological modeling. It is based on the transformation of the membrane potential into an instantaneous firing frequency, F , F =
fmax , 1 + exp(−ν(X − κ))
(6)
where ν and κ are parameters and fmax is the maximum achievable firing frequency (see Fig. 1). The advantage of this approach with respect to the previous one is obvious as it does not require one to solve the first-passage-time problem. Replacing X in (6) by the asymptotic mean of the membrane potential calculated from (2), E(X(∞)) = µ/α, we can identify the parameters ν and κ to make the curves (5) and (6) alike. If we require for both of them the same −1 ), the same location at which the curves reach the half-value of saturation (fmax /2) asymptotic value (fmax = tref and the same slope of the curves at this point, then κ=
S 1 − exp(−αtref )
(7)
ν=
(1 − exp(−αtref ))2 . αStref exp(−αtref )
(8)
and
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Fig. 1. The input–output transfer functions. In the left part of the figure are four intensity-frequency transfer functions for the single-point model. Among them, the full line illustrates relationship (5) with parameters S = 6 [mV], α = 1/3 [1/ms], tref = 2 [ms], the dot-dashed line corresponds to Eq. (6) with parameters κ and ν determined by (7) and (8) and X = µ/α. The remaining two irregular dashed lines are experimental counterparts of (5) calculated for average interspike intervals estimated from simulation (100 interspike intervals for each value of µ) of the stochastic single-point model (1) with the same parameters and σ = 3 [mV/ms1/2 ] (starts at −2) and σ = 6 [mV/ms1/2 ] for (starts at −5). In the right part of the figure are three intensity-frequency transfer functions for the two-point model with the same parameters as above and τr = 5. Here, the full line was obtained by numerical solution of (35) and the dashed lines were calculated for average interspike intervals estimated from simulation (again 100 interspike intervals per each µ) of the two-point model (9) and (10) with the same parameters as the curves in the left-hand part of the figure.
The main difference between curves (5) and (6) is at the initial phase, where (5) has a second order discontinuity at µ/α and its steepest slope is located at this point. Curve (6) is smooth and symmetrical with respect to µ/α = κ (see Fig. 1). This striking difference between the curves is partially removed by introducing stochasticity into the model (e.g. [31,32]). Very similar results to those derived from (6) can be obtained by using the function tanh, [33].
3. Two-point model In model (1) X represents the membrane depolarization in an abstract point, which is generally identified with the trigger zone. However, the input, which takes place mainly in the dendritic part of the neuron, is also represented here. This concentration of all properties into the single point is the dominant source of inadequacy of the model. Thus, in the two-point approach, the description by X is replaced by a couple X1 and X2 representing depolarizations in two distinct parts – dendritic and trigger zone compartments. In the previous study [11], the depolarization X1 of the dendritic compartment was assumed to be governed by the Ornstein–Uhlenbeck process (1). There, it followed from unidirectional connection between compartments, the axonal membrane potential X2 tracked exponentially the dendritic potential X1 , which evolved independently of X2 . If we remove the simplifying assumption that the dendritic compartment depolarization is not influenced by
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the voltage at the trigger zone, as done in [13] for the deterministic two-point model, then the new model can be formally defined in the following way: X1 (t) X2 (t) − X1 (t) + + µ(t) dt + σ (t)dW (9) dX1 (t) = − τ τr is the equation describing the dendritic potential X1 and for the potential X2 at the trigger zone holds X2 (t) X1 (t) − X2 (t) + dt, dX2 (t) = − τ τr
(10)
where τr is the junctional time constant, µ(t) represents the external input and σ (t) its variability. It is assumed in Eqs. (9) and (10) that the intrinsic parameters are the same in both compartments, τ = τ1 = τ2 , however, this simplification can be easily removed. In this article, we do not aim at direct description of the spiking activity by solving the first-passage-time problem for model (9) and (10). As mentioned before, the task is complicated even for the single-point model. Instead, we are mainly interested in the statistical moments of the subthreshold behavior, namely, the means, m1 (t) = E(X1 (t)) and m2 (t) = E(X2 (t)), and the second order moments. These quantities can be compared with (2) and (3). Further, the spiking activity can be related to the mean depolarization at the trigger zone, m2 , by using relation (6) and approximating the actual value of the potential, X2 by its mean. Variance (3) of the single point model has been used for checking the precision of the approximation of the firing frequency in the paper [34] and it can be expected that a similar approach will be applicable for the two-point model. A general formula for the moments of stochastic process given by linear stochastic differential equations can be applied on Eqs. (9) and (10) [35]. For the means we have dm1 (t) = −(α + αr )m1 (t) + αr m2 (t) + µ(t) dt
(11)
dm2 (t) = −(α + αr )m2 (t) + αr m1 (t), dt
(12)
and
where we denoted α = τ −1 and αr = τr−1 . The second non-central moments satisfy the following differential equations: σ 2 (t) 1 dE(X12 (t)) = −(α + αr )E(X12 (t)) + αr E(X1 (t)X2 (t)) + µ(t)m1 (t) + , 2 dt 2
(13)
1 dE(X22 (t)) = −(α + αr )E(X22 (t)) + αr E(X1 (t)X2 (t)) 2 dt
(14)
dE(X1 (t)X2 (t)) = −2(α + αr )E(X1 (t)X2 (t)) + αr (E(X1 (t)) + E(X2 (t))) + µ(t)m2 (t). dt
(15)
and
Solving the above equations, the variances of membrane depolarization at the dendritic and trigger zone compartments and covariance between them can be computed. The initial conditions for Eqs. (11)–(15) will be discussed in Section 4.
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4. Properties of the model 4.1. Subthreshold behavior For constant stimulation, µ(t) = µ, solutions of Eqs. (11)–(15) provide us with information analogous to (2)–(4). Due to constancy of µ, the time independent (homogeneous) variability of the input, σ (t) = σ is considered. The assumption of constant infinitesimal variance is legitimate for constant µ, but for time varying input, analogous variability of σ can be expected. Further, even for constant µ, the constant σ should be proportional to µ (see Section 5), however, to make our results comparable with previous works on the single-point model we use fixed σ over the complete range of µ. At first, we are interested in the steady-state behavior of the model. For this purpose we solve Eqs. (11)–(15) in which the left-hand sides are replaced by zeros. The solutions for (11) and (12) denoted by m1 (∞) and m2 (∞) are m1 (∞) =
(α + αr )µ α(α + 2αr )
(16)
m2 (∞) =
αr µ . α(α + 2αr )
(17)
and
We can see that m2 (∞) = m1 (∞) − µ/(α + 2αr ) and for αr = 0, which means that the two compartments are disconnected, (16) coincides with the limiting value of (2), m1 (∞) = E(X(∞)), and m2 (∞) = 0. With increasing value of αr , the difference between two mean depolarizations (16) and (17) decreases. This is well apparent from the ratio between m1 (∞) and m2 (∞), m1 (∞) = 1 + , m2 (∞)
(18)
where = α/αr and Eq. (18) also implies that |m2 (∞)| < |m1 (∞)| holds true. In the limiting situation αr α (the junctional time constant being much smaller than the transmembrane one), the depolarization is equally divided between both compartments, so m1 (∞) ≈ m2 (∞) ≈ E(X(∞))/2 = µ/(2α) holds. Further, we can see by comparing the asymptotic values of (2) and (17), that the asymptotic mean depolarization at the trigger zone of the two-point model is in absolute value lower than the asymptotic depolarization in the single-point model, if the same signal has been applied. For the ratio of these quantities we have E(X(∞)) = 2 + . m2 (∞)
(19)
The asymptotic second central moments are Var(X1 (∞)) =
(2α 2 + 4ααr + αr2 )σ 2 , 4α(α + αr )(α + 2αr )
(20)
Var(X2 (∞)) =
αr2 σ 2 4α(α + αr )(α + 2αr )
(21)
and Cov(X12 (∞)) =
αr σ 2 . 4α(α + 2αr )
(22)
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Fig. 2. Dependency of steady-state variances (3) and (21) of the single-and two-point models on the parameter α (the reciprocal of the time constant), for σ 2 = 1 and different values of junctional time constant αr−1 . The full line corresponds to the single-point model and the dashed lines to the two-point model from the top to the bottom with parameter αr = 10, 0.3 and 0.1 [1/ms].
So, the variance at the trigger zone compartment is always smaller than the variance at the dendritic compartment, Var(X2 (∞)) = Var(X1 (∞)) − σ 2 /[2(α + αr )], which is also well illustrated by the relationship Var(X1 (∞)) = 1 + 4 + 2 2 . Var(X2 (∞))
(23)
The dependency of Var(X2 (∞)) on α for different values of αr is illustrated in Fig. 2, from which we can see that whereas in the single-point model the variance decreases with 1/α, for the two-point model the decrease is much faster (proportional to 1/α 3 ). Like for the means, for αr = 0 variance (20) coincides with the limiting value of (3) and the remaining two central moments are equal to zero. We may also note that for αr > 0, variance (20) and (21) are always smaller than the asymptotic value of (3). Namely, for Var(X) and Var(X2 ) we have Var(X(∞)) = 4 + 12 + 2 2 , Var(X2 (∞))
(24)
from which follows that the variance at the spiking segment is at least four times higher in the single-point model than in the two-point one. In fact, appearing in (24) cannot be expected to be close to zero and thus this ratio is always much higher than the limiting minimum. Analogously to the single-point model, we can calculate the ratio between the asymptotic mean depolarization in the trigger zone compartment and its variance, ρ2 = |m2 (∞)|/Var(X2 (∞)). Using (17) and (21) we obtain ρ2 =
4|µ|(α + αr ) . σ 2 αr
(25)
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Fig. 3. Dependency of steady-state correlation coefficient (26), giving the correlation between depolarizations at the trigger zone and dendritic compartments, on the parameter αr . The different lines correspond to the values of parameter α = 10, 0.3 and 0.1 [1/ms] from bottom to the top.
By comparing (4) and (25) we see that ρ2 = 2ρ1 ((α + αr )/αr ) from where we deduce that ρ2 > ρ1 . For perfect connection between the compartments (αr → ∞) the quantity ρ2 is twice larger than ρ1 and this ratio grows with decreasing αr . From (20)–(22), we can calculate the asymptotic correlation coefficient α + αr Corr(X12 (∞)) = p 2 2α + 4ααr + αr2
(26)
which is always a positive and increasing √ function of αr and α, for which αr α implies Corr(X12 (∞)) → 1 while αr α induces Corr(X12 (∞)) → 1/ 2 (see Fig. 3). Note that for αr → 0, Corr(X12 ) has no practical meaning as X2 ∼ 0. Considering again the situation αr α, it is found that Var(X1 (α)) ≈ Var(X2 (∞)) ≈ Cov(X12 (∞)) ≈ Var(X(∞))/4 = σ 2 /8α, the total variance is equally divided among both compartments and covariance between them. The general solutions of (11) and (12) are µ 1 1 µ + m2 (0) − m1 (0) m1 (t)=m1 (∞)+ exp(−αt) − +m2 (0) + m1 (0) − exp(−(α+2αr )t) 2 α 2 α+2αr (27) and
µ 1 1 µ + m2 (0) − m1 (0) . m2 (t)=m2 (∞)+ exp(−αt) − +m2 (0)+m1 (0) + exp(−(α+2αr )t) 2 α 2 α+2αr (28)
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Of course, when comparing the means of the two-and single-point models, (27) and (28) with (2), we have E(X(t)) = m1 (t) + m2 (t). Further, one can write µ µ + exp(−(α + 2αr )t) + m2 (0) − m1 (0) , (29) m2 (t) = m1 (t) − α + 2αr α + 2αr from which it can be seen that the steady-state difference is achieved with time constant (α + 2αr )−1 . Finally, if we take as time origin the moment of spike generation, which implies that the initial depolarization of the axonal compartment is zero, X2 (0) = 0, and thus m2 (0) = 0 and if further it is assumed that the interspike intervals are sufficiently long for X1 to attain its steady-state level, m1 (0) = m1 (∞) then µ αr exp(−(α + 2αr )t) . (30) 1− m2 (t) = m1 (t) − α + 2αr α The general formulas for the second central moments are notationally complicated and we present only one simplified variant obtained for Var(X2 (0)) = 0, which corresponds to the fixed reset of the trigger zone potential, Var(X1 (t))=Var(X1 (∞))+
Var(X2 (t))=Var(X2 (∞))+
(1+e2αr t )2 Var(X1 (0)) 2αr2 e4αr t +ααr (1 + 4e2αr t +3e4αr t )+α 2 (1+e2αr t )2 2 − σ , 4e2(α+2αr )t 8α(α+αr )(α+2αr )e2(α+2αr )t (31) (e2αr t −1)2 Var(X1 (0)) 2αr2 e4αr t +ααr (1−4e2αr t +3e4αr t )+α 2 (e2αr t −1)2 2 − σ , 4e2(α+2αr )t 8α(α+αr )(α+2αr )e2(α+2αr )t (32)
and Cov(X12 (t)) = Cov(X12 (∞)) +
(e4αr t − 1)Var(X1 (0)) (α + 2αr )e4αr t − α 2 − σ . 4e2(α+2αr )t 8α(α + 2αr )e2(α+2αr )t
(33)
Combining (31) and (32), we obtain a formula analogous to (29) Var(X2 (t)) = Var(X1 (t)) −
σ2 σ2 Var(X1 (0)) , − 2(α+α )t + r 2(α + αr ) e 2(α + αr )e2(α+αr )t
(34)
which shows that the steady-state difference between the variances is achieved with time constant [2(α + αr )]−1 . Again, as for the first-order moments, comparing (3) with (31)–(33) we get Var(X(t)) = Var(X1 (t))+Var(X2 (t))+ 2Cov(X12 (t)). It can be seen from (24) or (34) that the variance of the depolarization at the trigger zone compartment is relatively small when compared with the single-point model (see Fig. 4, where averaged trajectories of X1 and X2 obtained from the simulation are presented). Further, employing the parameters used in Fig. 4, we can see that ρ2 ≈ 6ρ1 , which also illustrates the filtering effect of the dendritic compartment. 4.2. Threshold Before turning our attention to the problems of spiking in the two-point model, let us shortly mention the effect of the second compartment on the sigmoid frequency transfer function (6). The function is illustrated in Fig. 5 for both models using the asymptotic mean depolarizations instead of the actual voltage X. We can see that for µ = 0 the functions intersect at a single point (the same firing frequency is predicted by both models for a balanced input). Further, from the figure can be seen that introduction of the second compartment has a linearizing effect, which is pronounced for low levels of ατ .
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Fig. 4. Simulated two-point model under subthreshold stimulation. The average of X1 (t) and X2 (t), with parameters α = 0.33 [1/ms], αr = 0.2 [1/ms], σ = 1 [mV/ms1/2 ], µ = 1 [mV/ms] for t ∈ [35,125] and µ = 0 [mV/s], outside, calculated from 100 trials.
In accordance with the integrate-to-threshold scenario, in the moment when the membrane potential X2 at the trigger zone reaches the firing threshold S, the value of the process X2 is reset to zero while the process X1 continues in its evolution. If the time occurrence of this event is taken as the time origin, then for the complete description of the model we need to solve the first-passage-time problem for X2 under the initial conditions X1 (0) = x10 and X2 (t0 ) = 0, where x10 is the value of the dendritic potential at the time of the last spike. Non-reset of the dendritic potential X1 is the reason for the non-renewal character of firings produced by the two-compartment model, even for constant stimulation. The simulated system (9) and (10) with imposed firing threshold is depicted in Fig. 6. As follows from the picture as well as from the analytical results, the variabilities of the depolarizations X1 and X2 are not influenced by the stimulation intensity µ. For the selected set of parameters used in this illustration, the effect of the reset on the behavior of the dendritic compartment is not visible and it would be of interest to find a region of the parametric space where this influence is substantial. Especially, if we can assume that Var(X1 (0)) = Var(X1 (∞)) and m1 (0) = m1 (∞), then the approximation techniques for the evaluation of interspike intervals can be used, see below. The patterns of interspike intervals are clearly apparent from Fig. 6 and their serial dependency is documented by the scatter plot diagram, Fig. 7, which is a statistical technique often used in experimental studies. Even in the single-point model, it has been a common approach to approximate the length of interspike intervals by the time when the mean depolarization crosses the firing threshold. Let us employ the same technique here to find a relation between the intensity of stimulation and the firing frequency analogous to (5). For σ = 0, the Eqs. (27) and (28) describes the trajectories of the depolarizations in both compartments. For long-lasting constant stimulation, the regular firing is achieved under the condition that the firing threshold is lower than the asymptotic depolarization at the trigger zone compartment, S < m2 (∞) [13]. Then solving these two equations under the
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Fig. 5. The input–output transfer functions given by (6) for single-and two-point models. The depolarization X in (6) has been replaced by the asymptotic mean values E(X(∞)) and m2 (∞), which are given by (2) and (17). The parameters are the same as in Fig. 1 (S = 6 [mV], α = 1/3 [1/ms], tref = 2 [ms]). The full line corresponds to the single-point model, the dashed lines illustrates the input–output curves for the two-point model with the same parameters and with αr = 10, 0.3 and 0.1 [1/ms] from top to the bottom.
conditions m1 (tj ) = m1 (tj +1 ), m2 (tj ) = 0 and m2 (tj +1 ) = S, where tj = 0 and tj +1 are two consecutive instants of neuronal firing, we obtain an equation for the length of interspike intervals, s = tj +1 − tj , S − m2 (∞) = (e−αs + e−(α+2αr )s )
S − 2m2 (∞) + e−2(α+αr )s m2 (∞). 2
(35)
Relating 1/s to µ we produce the input–output curve analogous to those presented for the single-point model, see Fig. 1. For σ > 0, beside the simulation, for an approximation of the firing activity, we can employ the fact that the depolarization in the model is a two-dimensional Gaussian process. As a first approximation of the firing frequency, we can use the following approach. Due to the Gaussian character, X2 (t) is mostly concentrated within an envelope √ bounded by m2 (t) ± 2 Var(X2 (t)), where the quantities used are given by (28) and (32). Two distinct cases can be considered. (1) If m2 (∞) S, an approximate value of the time needed to cross the firing threshold is that at which m2 (t) reaches the threshold, and it is given by (35). For lower values of σ this approximation is better. One √ can expect that this approximation is good, if s1 and s2 obtained by replacing m2 (∞) by m2 (∞) ± 2 Var(X2 (∞)) do not differ substantially from s. The values of s, s1 and s2 can be found by solving (35). An application of this procedure is illustrated in Fig. 8 . It is apparent that for the selected parameters the fit between mean interspike interval predicted by this method and simulation is excellent. The fit of standard deviations does not seem to be so good, however, it has never been expected. (2) If m2 (∞) S, the firing threshold is crossed only due to the random fluctuations of the depolarization at the trigger zone compartment (which are highly reduced by the spatial properties of the neuron). Then, as in the single-point model, where the Poissonian character of the firing was proved analytically [36], also for the two-point model a tendency to Poissonian activity can be expected. Further, due to
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Fig. 6. Simulated two-point model with the firing threshold S = 1.6 [mV] imposed at the trigger zone compartment for stimulation intensity µ = 2 [mV/ms] for t ∈ [0,50], µ = 3 [mV/ms] for t ∈ [80,160] and µ = 0 [mV/ms] outside these intervals (a single trial). The other parameters are σ = 1 [mV/ms1/2 ], αr = 0.2 [1/ms], α = 0.33 [1/ms]. At the bottom the stimulation intervals are indicted. The reset of the trigger zone depolarizations X2 (t) at the moment of reaching the threshold is indicated and the interspike intervals correspond to the intervals between the consecutive resets.
the correlation structure in the depolarization, instead of single spikes, bursts of spikes will appear in accordance with a Poisson process. This effect was already observed in the simulated activity of the simplified two-point model [11]. Here, we simulated both types of models for increasing values of the threshold S. The Poissonian character is documented by tendency of interspike intervals to have the coefficient of variation (CV = standard deviation/mean) equal to one (Fig. 9). (It should be pointed that for large S, the simulation of the first-passage times becomes rather difficult [20].) Of course, due to the fact that m(∞) > m2 (∞), the Poissonian character does not appear for both models at the same thresholds. Whether the two-point model generates clusters or not depends on the parameters σ and αr and to solve this question would require extensive simulation studies.
5. Parameters of the model For any quantitative discussions about the models and their mutual comparison, as well as comparison with experimental data at least approximate information about their parameters is necessary. Despite the fact that the values of intrinsic parameters (r, R, C) employed in model (9) and (10) can be found in biophysical studies on neuronal membranes, we should not rely too much on them because the model is still too abstract. We have to realize that as the model has been constructed, the compartments are infinitely close one to another. It may mean that whereas τ can probably be directly identified with the transmembrane leakage, the parameter τr is more abstract and any direct conclusions about it would be premature. The most appropriate would be to estimate these parameters from the data recorded intracellularly accordingly to the model construction.
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Fig. 7. The scatter plot diagram (j th interspike interval, sj = tj +1 − tj , is plotted against (j − 1)th interval, sj −1 = tj − tj −1 ), for 500 interspike intervals obtained from simulation presented in the first part of Fig. 6 (µ = 2 [mV/ms]). The positive serial dependency of the interspike intervals is apparent.
The vague information about the parameters τ and τr is still more reliable than that on the input parameters µ and σ , about which we can only speculate. These discussions, in the single-point model, have usually been based on the Stein’s model because its parameters still keep their physical interpretation, that are the amplitudes of postsynaptic potentials (or their contribution to the membrane potential at the trigger zone) and the input frequencies [37]. Since for X1 an analogous diffusion approximation could be proposed as in the single-point model, we can use similar reasoning for very preliminary information about the input parameters in the two-point model. Let us remind, that following the diffusion approximation, we can assume µ ≈ fe a + fi i, σ 2 ≈ fe a 2 + fi i 2 , where fe fi are the excitatory an inhibitory input rates and a, i are the amplitudes of excitatory and inhibitory postsynaptic potentials. Only for illustration, let us mention the values used in [38,39], which are a = −i = 0.2 mV and spontaneous firing activity in cortex is about 1 Hz, evoked is about 10 Hz, which gives a range for fe and fi . If we suppose, for example, that one third of synapses are inhibitory, it would induce for spontaneous activity µ = 2 mV/ms, σ 2 = 1.2 mV2 /ms, which suggests that µ and σ should be of the same order. In general, we can find a relationship based on the above reasoning; if the sizes of excitatory and inhibitory postsynaptic potentials are the same (−i = a > 0), and q > 1 is the ratio between excitation and inhibition rates (fe = qfi ), then q −1 µ , = 2 (q + 1)a σ
(36)
which gives a preliminary hint about the magnitudes of µ and σ . It suggests that the ratio between µ and σ 2 is independent of the input frequencies under the condition that while changing the frequencies, their ratio fe /fi = q is kept constant. A result which is not qualitatively different is obtained by assuming that instead of the proportional increase of the input frequencies, the increase is due to the increase of fe exclusively. In conclusion, from (36) a
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Fig. 8. Approximation of the input–output transfer function for the two-point model and its precision. The full line in the middle illustrates the approximation obtained from (35) in the case of suprathreshold stimulation of the two-point model with parameters S = 6 [mV], α = 1/3 [1/ms], αr = 1/5 [1/ms], tref = 2 [ms], (a) σ = 2 [mV/ms1/2 ], (b) σ = 4 [mV/ms1/2 ], (c) σ = 8 [mV/ms1/2 ] and it is the same input–output transfer function as in the right-hand part of Fig. 1. The crosses along this line are the outcome of the model simulations (1000 trials). The remaining√two full lines correspond the intervals (further transformed to the frequency) obtained by the crossings of the threshold by the function m2 (t) ± Var(X2 (t)). The diamond-like symbols are results of the mentioned simulation in which the average interspike interval has been increased/decreased for its two empirical standard deviation. Apparently the fit of the limits becomes good only when the first-passage-time distribution starts to be symmetrical.
crude information can be deduced about the simultaneous changes of the input parameters due to an increasing strength of stimulation. Another information about the input parameters in the single-point Ornstein-Uhlenbeck model comes immediately from the experimental data [40], where direct estimation from interspike intervals has been performed. In their paper the estimated values of the parameter µ ranges from −6.77 to 3 and σ goes up to 15. 6. Discussion As shown, the two-point model is characterized by several features that distinguishes it from its single-point counterpart, namely: the delayed reaction to stimulation onset, the lower variability of the depolarization, the lower increase of the depolarization in dependency on the stimulus intensity and the correlated voltage between the compartments which is finally reflected by the correlation structure of the interspike intervals. Let us only remind, that this correlation structure of interspike intervals cannot be produced in the single-point model, unless it is sustained by some additional features, like for example, an inhibitory feedback. The delayed reaction, which is apparent from Fig. 4 and from comparison of Eq. (2) with (28), is caused by the transfer of the input signal from the dendrite to the trigger zone which is completely neglected in the single-point model. As a consequence, the two-point model neuron is more robust against “non significant” short changes in the stimulus intensity.
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Fig. 8. (Continued )
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Fig. 9. Dependency of coefficient of variation on the firing threshold in the simulated one-point model (crosses) and two-point model (diamonds) with parameters µ = 5 [mV/ms], α = 1/3 [1/ms], αr = 1/5 [1/ms], σ = 1 [mV/ms1/2 ]. Increasing the threshold, CV tends to one for both models. The asymptotic depolarization at the trigger zone are depicted, m2 (∞) = 4.09 [mV] and m(∞) = 15 [mV].
The lower variability of the voltage fluctuation due to the randomly varying input is illustrated in Fig. 2 and it appears again in Fig. 4 and 6. It follows from (24) that the standard deviation of the steady-state depolarization in the two-point model is at least twice smaller, but in reality much lower, than the corresponding value in the single-point model. Of course, this holds under the assumption that both models are exposed to the same level of noise. If we relate the variability of the depolarization to the variability of the interspike intervals, which is legitimate, then the two-point model responds to the variability of the input by lower variability of the interspike intervals than the one-point model, again it appears to be more robust to “random” perturbations in the signal. At the first glance, this partial filtering of the input variability may seem to have the disadvantage of decreasing the coding range of the neuron. We saw in Fig. 1 that increasing σ in the single-point model results in the increase of the coding range and introduction of the second compartment partially removes this advantage. However, this decrease of the coding range is entirely compensated by the increase described by Eq. (19) and illustrated in Fig. 5. Due to it, we can see that the increase of the steady-state mean of the two-point model is at least twice lower than that in the single-point model. Consequently, the coding range is at least twice as large in the two-point model as in the single-point one. These two opposite effects (decrease of variability and decrease of the proportionality growth constant) keep the coding range at least unchanged (if not increased). From the point of view of information transfer these seem to be positive features. The last effect of the simplest spatial characterization of a neuron is the mutual dependency between the depolarizations at different compartments (Fig. 3) and consequently the serial dependency of the interspike intervals illustrated in Fig. 7. This feature, discussed in detail in [11] for the incomplete two-point model, characterizes bursting/clustering activity so often observed experimentally [41–43]. The role of bursting activity for information transfer within the nervous system is apparently enormous (only an example is LTP/LTD induction studied recently
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in [44]) and the two-point model is the simplest possible neuronal model achieving bursting for time unstructured input. Despite the obvious biological relevance of the complete two-point approach, it seems that for constant input with low variability there is only a negligible difference between the model with bi-directional connection of the compartments [13] and the model in which only one-way voltage propagation is considered [11,12] (in Fig. 6 we see no effect upon X1 caused by the reset of X2 ). One of the reasons may be the fact, that none of these models describe the shape and the effects of the action potential which is here replaced by a mere post-firing reset of the voltage. If this simplification will be removed, then the backpropagation of the action potential into the dendrite will play its well recognized role [45]. However, this modification will simultaneously remove the appreciated simplicity of the model. On the other hand, for time dependent signals considered in [13] or for description of transient states with constant stimulation, the bi-directional connection of the compartments may appear as an important factor even in this simple integrate-and-fire model. The lower depolarization at the trigger zone than at the dendrite is an inherent characteristic of the model. One of the reasons is that the model analyzed throughout this paper does not consider the soma of the neuron as its specific part, where the contributions to the trigger zone membrane depolarization are of substantial size. Whereas the input at the dendrite causes small changes of the membrane potential and thus the system is well characterized by Eq. (9), it would be natural to expect that the incoming signal located at the soma has discontinuous effect on the depolarization [46,47]. Therefore, to system (9) and (10) a third equation describing the somatic compartment may be added and this would be characterized by discontinuous trajectories, X3 (t) 2X3 (t) − X2 (t) − X1 (t) (37) + dt + adN + + idN − , dX3 (t) = − τ τr where a > 0 and i < 0 represent amplitude of excitatory and inhibitory postsynaptic potentials of synapses located at the soma and N + and N − the corresponding (homogeneous or non homogeneous in dependency in the type of input) Poisson processes driving these synapses. Similarly, to include the intrinsic noise into the model we may further modify (10) into the form X2 (t) X3 (t) − X2 (t) (38) + dt + σ2 (t)dW2 , dX2 (t) = − τ τr where σ2 (t) measures the proportion of intrinsic variability which is reflected by independent Wiener process W2 . The equation for the depolarization at the dendritic compartment would remain unchanged, except the part describing the connection with the somatic compartment, X1 (t) X3 (t) − X1 (t) + µ(t) dt + σ1 (t)dW1 . (39) + dX1 (t) = − τ τr These two modifications will change the proportion of the voltages at the different compartments and will be reconsidered in a forthcoming paper.
Acknowledgements We thank J.-P. Rospars for helpful discussions during the preparation of the paper. This work was supported by Academy of Sciences of the Czech Republic Grant No. A7011712/1997 and by GIS “Science de la Congnition” Grant CNA 10.
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Two-compartment stochastic model of a neuron Petr Lánský a,∗ , Roger Rodriguez b a
b
Institute of Physiology, Academy of Sciences of the Czech Republic, Videñská 1083, 142 20 Prague 4, Czech Republic Centre de Physique Théorique, CNRS and Faculté des Sciences de Luminy, Université de la Méditerranée, Luminy-Case 907, F-13288 Marseille Cedex 09, France Received 11 August 1998; received in revised form 7 February 1999; accepted 12 February 1999 Communicated by A.M. Albano
Abstract A two-compartment neuronal model is investigated. The model neuron is composed of two interconnected parts – a dendrite and a trigger zone, with a white noise input in the dendritic compartment. The first and the second moments of the stochastic processes describing the membrane depolarization in both compartments are derived and investigated. When a firing threshold is not imposed, the level of neuronal activity is deduced from a generally accepted relationship between the membrane potential and the firing frequency. When a firing threshold is imposed, some approximations and simulations are used to characterize the spiking activity of the model. It is shown that the activity of the two-compartment model is less sensitive to abrupt changes in stimulation than the activity of the single-compartment model. The delayed response of the complex model is a natural consequence of the fact that the input takes place in the compartment different from that at which the output is generated. Further, the model predicts serial correlation of interspike intervals, which is a phenomenon often observed in experimental data but not reproducible in the single-compartment models under steady-state stimulation. Finally, the investigated model neuron shows lower sensitivity to the input intensity and larger coding range than the single-compartment model. ©1999 Elsevier Science B.V. All rights reserved. PACS: 87.10.+e; 05.40.+j Keywords: Neuron model; Integrate-and-fire; Stochastic diffusion process; Membrane depolarization; Bursting
1. Introduction There is an enormously wide range of approaches to the modeling of neuronal activity starting with deterministic biophysical concepts which have proved to be very successful in explaining the generation of the various types of membrane potentials, such as the Hodgkin–Huxley model of the action potential. On the other end of the spectrum, if considering the integrated activity of single neurons or assemblies of neurons, some drastic simplifications are necessary. The leaky integrate-and-fire, as the most simplified but still sufficiently realistic model of the Hodgkin–Huxley schema, appears to be the most suitable for this purpose [1,2]. However, both of these two very ∗
Corresponding author. Tel.: +42-02-475 25 85; fax: +42-02-475 24 88; e-mail: [email protected].
0167-2789/99/$ – see front matter ©1999 Elsevier Science B.V. All rights reserved. PII: S 0 1 6 7 - 2 7 8 9 ( 9 9 ) 0 0 0 3 4 - 2
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different approaches have in common the fact that all the properties of the model neurons are concentrated into a single point in space (single-point models) and no geometrical architecture of the cell is considered. In contrast to this neuron representation are computerized multicompartment models (for a review see [3,4]) or cable models, either deterministic (for a review see [5]) or stochastic [6,7]. These spatially complex models usually do not aim at direct description of the input–output transfer but are mainly focused on the computational aspects of the information processing within the neuron itself. Besides simplifying the properties of the membrane along the way from Hodgkin–Huxley model to the leakyintegrator, and transforming the infinitely variable spatial structure into a single point, another set of assumptions deals with the properties of the neuronal input. An apparent variability in neuronal activity stresses the fact that if input–output transfer of a neuron or of a neuronal system is investigated, the stochastic phenomena may play an important role. The sources of randomness can be intrinsic as well as extrinsic. Nevertheless, it seems to be more likely that the stochastic variation of the input causes global changes and is crucial for the modification of the input–output information transfer characteristics [8]. The simplest approach to introduce external random influence into the neuronal models is considering a stochastic variant of the leaky integrate-and-fire model, which is based on two assumptions describing its behavior between two consecutive firings (resets). Firstly, the mean value of the membrane depolarization is described by the deterministic leaky integrate-and-fire model; secondly, the membrane potential at any given time instant is normally distributed around its mean value. The stochastic process fulfilling these requirements is an Ornstein–Uhlenbeck diffusion process. Stochastic models based on the leaky integrate-and-fire concept has been extensively used, modified and generalized (for a review see [9]). As stressed above, the tractability of these models follows from the fact that all the properties of the neuron are concentrated into a single point in space with none of the geometrical properties of a real neuron being taken into account, and it is a substantial simplification of reality. Therefore, in the recent years, several attempts appeared to generalize the single-point models making them more biologically relevant but still mathematically tractable. Kohn [10] proposed a two-compartment (two-point) model and similar models were further developed and studied [11,12]. The model we analyze here belongs to this class and it is based on the following set of hypotheses [13]: (i) The neuron is assumed to be made of two interconnected – dendritic and trigger zone – compartments. (ii) The input is present at the dendritic compartment only. (iii) The potentials of the two compartments are described by leaky integrators with a reset mechanism at the trigger zone. It follows from (i) that in this two-point model the dendritic potential depends on the trigger zone potential, a feature which was for the sake of tractability neglected in our previous papers [11,12]. The aim of this article is to investigate the first two statistical moments of the membrane depolarization in both compartments and to find their implications for the coding properties of the model. A comparison is made with activity evoked in the single-point model being stimulated by the same signal – constant or square pulse stimulation.
2. Single-point model Before going into the details about the two-point model, let us summarize the relevant information on its singlepoint counterpart (for a simple introduction to the single-point models see [14], a complete review can be found in [9]). One of the most common description of the stochastic evolution of the membrane potential under the singlepoint scenario is Stein’s model [15], which employs a Poissonian character of the timing of the incoming signals. It is an easy-to-interpret starting point from which more abstract Ornstein–Uhlenbeck model can be meaningfully derived (e.g. [16,17]). Despite that one could arrive at the stochastic leaky integrator just by adding the white noise
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to the input of the deterministic model, the approach via the Stein’s model is more illustrative and suitable for parameter interpretation (see Section 5). 2.1. Subthreshold behavior In the Ornstein–Uhlenbeck model, the behavior of the depolarization X of the membrane is described by the stochastic differential equation X (1) X(0) = X0 , dX = − + µ dt + σ dW, τ where W is a standard Wiener process, τ > 0 is the membrane time constant (τ = RC, where R is the membrane resistance and C is its capacitance), µ and σ > 0 are constants. The parameters µ and σ 2 in (1), expressed respectively in mV and mV2 per unit time, reflect the input signal and its variability resulting from the stochastic dendritic currents generated by sensory stimulation or by action of other neurons. The random initial depolarization defined by X0 was studied in [18] and recently in [19], but commonly it is assumed that Prob(X0 = 0) = 1. A direct generalization of model (1) assumes that parameters µ and σ 2 depend on time, reflecting time course of the signal impinging on the neuron (for discussion see [20]). Namely, constant σ and µ(t) = µ0 + µ1 cos(ωt), in addition to the reset after firing, is the most common generalization. This system produces stochastic resonance effects [21,22], a phenomenon which was expected and has been extensively searched for in the nervous system in the recent years (e.g. [23–25]). The mean and variance of X given by (1) are µ µ exp(−αt), (2) E(X(t)) = + E(X0 ) − α α σ2 σ2 + Var(X0 ) − exp(−2αt), (3) Var(X(t)) = 2α 2α where α = τ −1 . These quantities describe completely the behavior of the membrane depolarization because process (1) is Gaussian. Thus, from (2) and (3), at the steady-state, X(∞) ∼ N (µ/α, σ 2 /2α). From Eqs. (2) and (3) the relation between E(X(t)) and Var(X(t)) can be calculated. Denoting the asymptotic ratio between the absolute value of the mean depolarization and its variance by ρ1 , ρ1 = |E(X(∞))|/Var(X(∞)), we have ρ1 =
2|µ| σ2
(4)
and this value will be compared with an analogous one for the two-point model to show improvement in the signal-to-noise ratio achieved by introducing the second compartment. 2.2. Threshold Single-point model (1) is only appropriate for subthreshold response. If the neuron is capable of generating spike, a threshold condition has to be imposed. The firing of an action potential is identified with the first crossing of a firing threshold S, S > X0 and the time origin is the moment of the last firing. At the moments of spike generation the membrane potential is repeatedly reset to its initial value X0 . The reset of X after firing introduces a strong nonlinearity into the model. Further, it causes that no information about the input prior to firing prevails on the neuron. Therefore the interspike intervals are described by independent realizations of the random variable T = inf{t > 0; X(t) ≥ S|X0 < S}.
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Under the frequency coding scenario, which is plausible for constant and long-lasting stimulations, the shapes of the input–output frequency curves are used to characterize the models as well as the experimental data. In the latter case, the curve is obtained by plotting 1/t¯ versus the stimulus intensity, where t¯ is the average interspike interval. For the deterministic version of model (1), σ = 0, the length of the interspike interval can be easily calculated. It is (1/α) ln(µ/(µ − αS)) and the shape of the transfer function follows directly from it, ( −1 µ 1 ln , µ/α > S, + t ref α µ−αS (5) f (µ) = 0, µ/α ≤ S, where tref denotes an absolute refractory period preventing permanent increase of f with increasing µ, namely −1 in (5) µ → ∞ implies f → fmax = tref . Relating the maximum frequency to the absolute refractory period is only formal because in reality the saturation frequency limiting the activity of a neuron comes earlier for complex biophysical reasons. For σ > 0, the solution of the first-passage-time problem is not a simple task even for apparently simply looking model (1) and therefore numerical (e.g. [26,27]) and simulation [28] techniques have often been proposed. The shapes of the input–output frequency curves for model (1) can be found in [11], where the simulated first-passage times were treated in the same way as experimentally measured interspike intervals, which means that E(T ) was estimated by averaging over many realizations of the times when the simulated trajectories of X(t) crossed the firing threshold for the first time. Similarly, applying the numerical techniques, the curves are constructed by plotting 1/(E(T ) + tref ), where the mean first-passage time is calculated from available theoretical formulas [29,26]. Defining the coding range of a neuron, and similarly of a model neuron, as a set of µ which are characterized by different firing frequencies [30], the coding range of the deterministic counterpart of model (1) is enlarged by adding the noise. The firing activity of the noisy model starts at the levels below µ/α which is required by condition (5) for the deterministic model. In general, increasing the amplitude of the noise (σ in (1)), the transfer function gets broader and thus, in its left-hand part, becomes closer to the sigmoid function. 2.3. Sigmoid transfer function The above introduced transfer function (5) may be related to another one which is more common in neural network applications than in biological modeling. It is based on the transformation of the membrane potential into an instantaneous firing frequency, F , F =
fmax , 1 + exp(−ν(X − κ))
(6)
where ν and κ are parameters and fmax is the maximum achievable firing frequency (see Fig. 1). The advantage of this approach with respect to the previous one is obvious as it does not require one to solve the first-passage-time problem. Replacing X in (6) by the asymptotic mean of the membrane potential calculated from (2), E(X(∞)) = µ/α, we can identify the parameters ν and κ to make the curves (5) and (6) alike. If we require for both of them the same −1 ), the same location at which the curves reach the half-value of saturation (fmax /2) asymptotic value (fmax = tref and the same slope of the curves at this point, then κ=
S 1 − exp(−αtref )
(7)
ν=
(1 − exp(−αtref ))2 . αStref exp(−αtref )
(8)
and
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Fig. 1. The input–output transfer functions. In the left part of the figure are four intensity-frequency transfer functions for the single-point model. Among them, the full line illustrates relationship (5) with parameters S = 6 [mV], α = 1/3 [1/ms], tref = 2 [ms], the dot-dashed line corresponds to Eq. (6) with parameters κ and ν determined by (7) and (8) and X = µ/α. The remaining two irregular dashed lines are experimental counterparts of (5) calculated for average interspike intervals estimated from simulation (100 interspike intervals for each value of µ) of the stochastic single-point model (1) with the same parameters and σ = 3 [mV/ms1/2 ] (starts at −2) and σ = 6 [mV/ms1/2 ] for (starts at −5). In the right part of the figure are three intensity-frequency transfer functions for the two-point model with the same parameters as above and τr = 5. Here, the full line was obtained by numerical solution of (35) and the dashed lines were calculated for average interspike intervals estimated from simulation (again 100 interspike intervals per each µ) of the two-point model (9) and (10) with the same parameters as the curves in the left-hand part of the figure.
The main difference between curves (5) and (6) is at the initial phase, where (5) has a second order discontinuity at µ/α and its steepest slope is located at this point. Curve (6) is smooth and symmetrical with respect to µ/α = κ (see Fig. 1). This striking difference between the curves is partially removed by introducing stochasticity into the model (e.g. [31,32]). Very similar results to those derived from (6) can be obtained by using the function tanh, [33].
3. Two-point model In model (1) X represents the membrane depolarization in an abstract point, which is generally identified with the trigger zone. However, the input, which takes place mainly in the dendritic part of the neuron, is also represented here. This concentration of all properties into the single point is the dominant source of inadequacy of the model. Thus, in the two-point approach, the description by X is replaced by a couple X1 and X2 representing depolarizations in two distinct parts – dendritic and trigger zone compartments. In the previous study [11], the depolarization X1 of the dendritic compartment was assumed to be governed by the Ornstein–Uhlenbeck process (1). There, it followed from unidirectional connection between compartments, the axonal membrane potential X2 tracked exponentially the dendritic potential X1 , which evolved independently of X2 . If we remove the simplifying assumption that the dendritic compartment depolarization is not influenced by
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the voltage at the trigger zone, as done in [13] for the deterministic two-point model, then the new model can be formally defined in the following way: X1 (t) X2 (t) − X1 (t) + + µ(t) dt + σ (t)dW (9) dX1 (t) = − τ τr is the equation describing the dendritic potential X1 and for the potential X2 at the trigger zone holds X2 (t) X1 (t) − X2 (t) + dt, dX2 (t) = − τ τr
(10)
where τr is the junctional time constant, µ(t) represents the external input and σ (t) its variability. It is assumed in Eqs. (9) and (10) that the intrinsic parameters are the same in both compartments, τ = τ1 = τ2 , however, this simplification can be easily removed. In this article, we do not aim at direct description of the spiking activity by solving the first-passage-time problem for model (9) and (10). As mentioned before, the task is complicated even for the single-point model. Instead, we are mainly interested in the statistical moments of the subthreshold behavior, namely, the means, m1 (t) = E(X1 (t)) and m2 (t) = E(X2 (t)), and the second order moments. These quantities can be compared with (2) and (3). Further, the spiking activity can be related to the mean depolarization at the trigger zone, m2 , by using relation (6) and approximating the actual value of the potential, X2 by its mean. Variance (3) of the single point model has been used for checking the precision of the approximation of the firing frequency in the paper [34] and it can be expected that a similar approach will be applicable for the two-point model. A general formula for the moments of stochastic process given by linear stochastic differential equations can be applied on Eqs. (9) and (10) [35]. For the means we have dm1 (t) = −(α + αr )m1 (t) + αr m2 (t) + µ(t) dt
(11)
dm2 (t) = −(α + αr )m2 (t) + αr m1 (t), dt
(12)
and
where we denoted α = τ −1 and αr = τr−1 . The second non-central moments satisfy the following differential equations: σ 2 (t) 1 dE(X12 (t)) = −(α + αr )E(X12 (t)) + αr E(X1 (t)X2 (t)) + µ(t)m1 (t) + , 2 dt 2
(13)
1 dE(X22 (t)) = −(α + αr )E(X22 (t)) + αr E(X1 (t)X2 (t)) 2 dt
(14)
dE(X1 (t)X2 (t)) = −2(α + αr )E(X1 (t)X2 (t)) + αr (E(X1 (t)) + E(X2 (t))) + µ(t)m2 (t). dt
(15)
and
Solving the above equations, the variances of membrane depolarization at the dendritic and trigger zone compartments and covariance between them can be computed. The initial conditions for Eqs. (11)–(15) will be discussed in Section 4.
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4. Properties of the model 4.1. Subthreshold behavior For constant stimulation, µ(t) = µ, solutions of Eqs. (11)–(15) provide us with information analogous to (2)–(4). Due to constancy of µ, the time independent (homogeneous) variability of the input, σ (t) = σ is considered. The assumption of constant infinitesimal variance is legitimate for constant µ, but for time varying input, analogous variability of σ can be expected. Further, even for constant µ, the constant σ should be proportional to µ (see Section 5), however, to make our results comparable with previous works on the single-point model we use fixed σ over the complete range of µ. At first, we are interested in the steady-state behavior of the model. For this purpose we solve Eqs. (11)–(15) in which the left-hand sides are replaced by zeros. The solutions for (11) and (12) denoted by m1 (∞) and m2 (∞) are m1 (∞) =
(α + αr )µ α(α + 2αr )
(16)
m2 (∞) =
αr µ . α(α + 2αr )
(17)
and
We can see that m2 (∞) = m1 (∞) − µ/(α + 2αr ) and for αr = 0, which means that the two compartments are disconnected, (16) coincides with the limiting value of (2), m1 (∞) = E(X(∞)), and m2 (∞) = 0. With increasing value of αr , the difference between two mean depolarizations (16) and (17) decreases. This is well apparent from the ratio between m1 (∞) and m2 (∞), m1 (∞) = 1 + , m2 (∞)
(18)
where = α/αr and Eq. (18) also implies that |m2 (∞)| < |m1 (∞)| holds true. In the limiting situation αr α (the junctional time constant being much smaller than the transmembrane one), the depolarization is equally divided between both compartments, so m1 (∞) ≈ m2 (∞) ≈ E(X(∞))/2 = µ/(2α) holds. Further, we can see by comparing the asymptotic values of (2) and (17), that the asymptotic mean depolarization at the trigger zone of the two-point model is in absolute value lower than the asymptotic depolarization in the single-point model, if the same signal has been applied. For the ratio of these quantities we have E(X(∞)) = 2 + . m2 (∞)
(19)
The asymptotic second central moments are Var(X1 (∞)) =
(2α 2 + 4ααr + αr2 )σ 2 , 4α(α + αr )(α + 2αr )
(20)
Var(X2 (∞)) =
αr2 σ 2 4α(α + αr )(α + 2αr )
(21)
and Cov(X12 (∞)) =
αr σ 2 . 4α(α + 2αr )
(22)
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Fig. 2. Dependency of steady-state variances (3) and (21) of the single-and two-point models on the parameter α (the reciprocal of the time constant), for σ 2 = 1 and different values of junctional time constant αr−1 . The full line corresponds to the single-point model and the dashed lines to the two-point model from the top to the bottom with parameter αr = 10, 0.3 and 0.1 [1/ms].
So, the variance at the trigger zone compartment is always smaller than the variance at the dendritic compartment, Var(X2 (∞)) = Var(X1 (∞)) − σ 2 /[2(α + αr )], which is also well illustrated by the relationship Var(X1 (∞)) = 1 + 4 + 2 2 . Var(X2 (∞))
(23)
The dependency of Var(X2 (∞)) on α for different values of αr is illustrated in Fig. 2, from which we can see that whereas in the single-point model the variance decreases with 1/α, for the two-point model the decrease is much faster (proportional to 1/α 3 ). Like for the means, for αr = 0 variance (20) coincides with the limiting value of (3) and the remaining two central moments are equal to zero. We may also note that for αr > 0, variance (20) and (21) are always smaller than the asymptotic value of (3). Namely, for Var(X) and Var(X2 ) we have Var(X(∞)) = 4 + 12 + 2 2 , Var(X2 (∞))
(24)
from which follows that the variance at the spiking segment is at least four times higher in the single-point model than in the two-point one. In fact, appearing in (24) cannot be expected to be close to zero and thus this ratio is always much higher than the limiting minimum. Analogously to the single-point model, we can calculate the ratio between the asymptotic mean depolarization in the trigger zone compartment and its variance, ρ2 = |m2 (∞)|/Var(X2 (∞)). Using (17) and (21) we obtain ρ2 =
4|µ|(α + αr ) . σ 2 αr
(25)
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Fig. 3. Dependency of steady-state correlation coefficient (26), giving the correlation between depolarizations at the trigger zone and dendritic compartments, on the parameter αr . The different lines correspond to the values of parameter α = 10, 0.3 and 0.1 [1/ms] from bottom to the top.
By comparing (4) and (25) we see that ρ2 = 2ρ1 ((α + αr )/αr ) from where we deduce that ρ2 > ρ1 . For perfect connection between the compartments (αr → ∞) the quantity ρ2 is twice larger than ρ1 and this ratio grows with decreasing αr . From (20)–(22), we can calculate the asymptotic correlation coefficient α + αr Corr(X12 (∞)) = p 2 2α + 4ααr + αr2
(26)
which is always a positive and increasing √ function of αr and α, for which αr α implies Corr(X12 (∞)) → 1 while αr α induces Corr(X12 (∞)) → 1/ 2 (see Fig. 3). Note that for αr → 0, Corr(X12 ) has no practical meaning as X2 ∼ 0. Considering again the situation αr α, it is found that Var(X1 (α)) ≈ Var(X2 (∞)) ≈ Cov(X12 (∞)) ≈ Var(X(∞))/4 = σ 2 /8α, the total variance is equally divided among both compartments and covariance between them. The general solutions of (11) and (12) are µ 1 1 µ + m2 (0) − m1 (0) m1 (t)=m1 (∞)+ exp(−αt) − +m2 (0) + m1 (0) − exp(−(α+2αr )t) 2 α 2 α+2αr (27) and
µ 1 1 µ + m2 (0) − m1 (0) . m2 (t)=m2 (∞)+ exp(−αt) − +m2 (0)+m1 (0) + exp(−(α+2αr )t) 2 α 2 α+2αr (28)
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Of course, when comparing the means of the two-and single-point models, (27) and (28) with (2), we have E(X(t)) = m1 (t) + m2 (t). Further, one can write µ µ + exp(−(α + 2αr )t) + m2 (0) − m1 (0) , (29) m2 (t) = m1 (t) − α + 2αr α + 2αr from which it can be seen that the steady-state difference is achieved with time constant (α + 2αr )−1 . Finally, if we take as time origin the moment of spike generation, which implies that the initial depolarization of the axonal compartment is zero, X2 (0) = 0, and thus m2 (0) = 0 and if further it is assumed that the interspike intervals are sufficiently long for X1 to attain its steady-state level, m1 (0) = m1 (∞) then µ αr exp(−(α + 2αr )t) . (30) 1− m2 (t) = m1 (t) − α + 2αr α The general formulas for the second central moments are notationally complicated and we present only one simplified variant obtained for Var(X2 (0)) = 0, which corresponds to the fixed reset of the trigger zone potential, Var(X1 (t))=Var(X1 (∞))+
Var(X2 (t))=Var(X2 (∞))+
(1+e2αr t )2 Var(X1 (0)) 2αr2 e4αr t +ααr (1 + 4e2αr t +3e4αr t )+α 2 (1+e2αr t )2 2 − σ , 4e2(α+2αr )t 8α(α+αr )(α+2αr )e2(α+2αr )t (31) (e2αr t −1)2 Var(X1 (0)) 2αr2 e4αr t +ααr (1−4e2αr t +3e4αr t )+α 2 (e2αr t −1)2 2 − σ , 4e2(α+2αr )t 8α(α+αr )(α+2αr )e2(α+2αr )t (32)
and Cov(X12 (t)) = Cov(X12 (∞)) +
(e4αr t − 1)Var(X1 (0)) (α + 2αr )e4αr t − α 2 − σ . 4e2(α+2αr )t 8α(α + 2αr )e2(α+2αr )t
(33)
Combining (31) and (32), we obtain a formula analogous to (29) Var(X2 (t)) = Var(X1 (t)) −
σ2 σ2 Var(X1 (0)) , − 2(α+α )t + r 2(α + αr ) e 2(α + αr )e2(α+αr )t
(34)
which shows that the steady-state difference between the variances is achieved with time constant [2(α + αr )]−1 . Again, as for the first-order moments, comparing (3) with (31)–(33) we get Var(X(t)) = Var(X1 (t))+Var(X2 (t))+ 2Cov(X12 (t)). It can be seen from (24) or (34) that the variance of the depolarization at the trigger zone compartment is relatively small when compared with the single-point model (see Fig. 4, where averaged trajectories of X1 and X2 obtained from the simulation are presented). Further, employing the parameters used in Fig. 4, we can see that ρ2 ≈ 6ρ1 , which also illustrates the filtering effect of the dendritic compartment. 4.2. Threshold Before turning our attention to the problems of spiking in the two-point model, let us shortly mention the effect of the second compartment on the sigmoid frequency transfer function (6). The function is illustrated in Fig. 5 for both models using the asymptotic mean depolarizations instead of the actual voltage X. We can see that for µ = 0 the functions intersect at a single point (the same firing frequency is predicted by both models for a balanced input). Further, from the figure can be seen that introduction of the second compartment has a linearizing effect, which is pronounced for low levels of ατ .
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Fig. 4. Simulated two-point model under subthreshold stimulation. The average of X1 (t) and X2 (t), with parameters α = 0.33 [1/ms], αr = 0.2 [1/ms], σ = 1 [mV/ms1/2 ], µ = 1 [mV/ms] for t ∈ [35,125] and µ = 0 [mV/s], outside, calculated from 100 trials.
In accordance with the integrate-to-threshold scenario, in the moment when the membrane potential X2 at the trigger zone reaches the firing threshold S, the value of the process X2 is reset to zero while the process X1 continues in its evolution. If the time occurrence of this event is taken as the time origin, then for the complete description of the model we need to solve the first-passage-time problem for X2 under the initial conditions X1 (0) = x10 and X2 (t0 ) = 0, where x10 is the value of the dendritic potential at the time of the last spike. Non-reset of the dendritic potential X1 is the reason for the non-renewal character of firings produced by the two-compartment model, even for constant stimulation. The simulated system (9) and (10) with imposed firing threshold is depicted in Fig. 6. As follows from the picture as well as from the analytical results, the variabilities of the depolarizations X1 and X2 are not influenced by the stimulation intensity µ. For the selected set of parameters used in this illustration, the effect of the reset on the behavior of the dendritic compartment is not visible and it would be of interest to find a region of the parametric space where this influence is substantial. Especially, if we can assume that Var(X1 (0)) = Var(X1 (∞)) and m1 (0) = m1 (∞), then the approximation techniques for the evaluation of interspike intervals can be used, see below. The patterns of interspike intervals are clearly apparent from Fig. 6 and their serial dependency is documented by the scatter plot diagram, Fig. 7, which is a statistical technique often used in experimental studies. Even in the single-point model, it has been a common approach to approximate the length of interspike intervals by the time when the mean depolarization crosses the firing threshold. Let us employ the same technique here to find a relation between the intensity of stimulation and the firing frequency analogous to (5). For σ = 0, the Eqs. (27) and (28) describes the trajectories of the depolarizations in both compartments. For long-lasting constant stimulation, the regular firing is achieved under the condition that the firing threshold is lower than the asymptotic depolarization at the trigger zone compartment, S < m2 (∞) [13]. Then solving these two equations under the
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Fig. 5. The input–output transfer functions given by (6) for single-and two-point models. The depolarization X in (6) has been replaced by the asymptotic mean values E(X(∞)) and m2 (∞), which are given by (2) and (17). The parameters are the same as in Fig. 1 (S = 6 [mV], α = 1/3 [1/ms], tref = 2 [ms]). The full line corresponds to the single-point model, the dashed lines illustrates the input–output curves for the two-point model with the same parameters and with αr = 10, 0.3 and 0.1 [1/ms] from top to the bottom.
conditions m1 (tj ) = m1 (tj +1 ), m2 (tj ) = 0 and m2 (tj +1 ) = S, where tj = 0 and tj +1 are two consecutive instants of neuronal firing, we obtain an equation for the length of interspike intervals, s = tj +1 − tj , S − m2 (∞) = (e−αs + e−(α+2αr )s )
S − 2m2 (∞) + e−2(α+αr )s m2 (∞). 2
(35)
Relating 1/s to µ we produce the input–output curve analogous to those presented for the single-point model, see Fig. 1. For σ > 0, beside the simulation, for an approximation of the firing activity, we can employ the fact that the depolarization in the model is a two-dimensional Gaussian process. As a first approximation of the firing frequency, we can use the following approach. Due to the Gaussian character, X2 (t) is mostly concentrated within an envelope √ bounded by m2 (t) ± 2 Var(X2 (t)), where the quantities used are given by (28) and (32). Two distinct cases can be considered. (1) If m2 (∞) S, an approximate value of the time needed to cross the firing threshold is that at which m2 (t) reaches the threshold, and it is given by (35). For lower values of σ this approximation is better. One √ can expect that this approximation is good, if s1 and s2 obtained by replacing m2 (∞) by m2 (∞) ± 2 Var(X2 (∞)) do not differ substantially from s. The values of s, s1 and s2 can be found by solving (35). An application of this procedure is illustrated in Fig. 8 . It is apparent that for the selected parameters the fit between mean interspike interval predicted by this method and simulation is excellent. The fit of standard deviations does not seem to be so good, however, it has never been expected. (2) If m2 (∞) S, the firing threshold is crossed only due to the random fluctuations of the depolarization at the trigger zone compartment (which are highly reduced by the spatial properties of the neuron). Then, as in the single-point model, where the Poissonian character of the firing was proved analytically [36], also for the two-point model a tendency to Poissonian activity can be expected. Further, due to
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Fig. 6. Simulated two-point model with the firing threshold S = 1.6 [mV] imposed at the trigger zone compartment for stimulation intensity µ = 2 [mV/ms] for t ∈ [0,50], µ = 3 [mV/ms] for t ∈ [80,160] and µ = 0 [mV/ms] outside these intervals (a single trial). The other parameters are σ = 1 [mV/ms1/2 ], αr = 0.2 [1/ms], α = 0.33 [1/ms]. At the bottom the stimulation intervals are indicted. The reset of the trigger zone depolarizations X2 (t) at the moment of reaching the threshold is indicated and the interspike intervals correspond to the intervals between the consecutive resets.
the correlation structure in the depolarization, instead of single spikes, bursts of spikes will appear in accordance with a Poisson process. This effect was already observed in the simulated activity of the simplified two-point model [11]. Here, we simulated both types of models for increasing values of the threshold S. The Poissonian character is documented by tendency of interspike intervals to have the coefficient of variation (CV = standard deviation/mean) equal to one (Fig. 9). (It should be pointed that for large S, the simulation of the first-passage times becomes rather difficult [20].) Of course, due to the fact that m(∞) > m2 (∞), the Poissonian character does not appear for both models at the same thresholds. Whether the two-point model generates clusters or not depends on the parameters σ and αr and to solve this question would require extensive simulation studies.
5. Parameters of the model For any quantitative discussions about the models and their mutual comparison, as well as comparison with experimental data at least approximate information about their parameters is necessary. Despite the fact that the values of intrinsic parameters (r, R, C) employed in model (9) and (10) can be found in biophysical studies on neuronal membranes, we should not rely too much on them because the model is still too abstract. We have to realize that as the model has been constructed, the compartments are infinitely close one to another. It may mean that whereas τ can probably be directly identified with the transmembrane leakage, the parameter τr is more abstract and any direct conclusions about it would be premature. The most appropriate would be to estimate these parameters from the data recorded intracellularly accordingly to the model construction.
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Fig. 7. The scatter plot diagram (j th interspike interval, sj = tj +1 − tj , is plotted against (j − 1)th interval, sj −1 = tj − tj −1 ), for 500 interspike intervals obtained from simulation presented in the first part of Fig. 6 (µ = 2 [mV/ms]). The positive serial dependency of the interspike intervals is apparent.
The vague information about the parameters τ and τr is still more reliable than that on the input parameters µ and σ , about which we can only speculate. These discussions, in the single-point model, have usually been based on the Stein’s model because its parameters still keep their physical interpretation, that are the amplitudes of postsynaptic potentials (or their contribution to the membrane potential at the trigger zone) and the input frequencies [37]. Since for X1 an analogous diffusion approximation could be proposed as in the single-point model, we can use similar reasoning for very preliminary information about the input parameters in the two-point model. Let us remind, that following the diffusion approximation, we can assume µ ≈ fe a + fi i, σ 2 ≈ fe a 2 + fi i 2 , where fe fi are the excitatory an inhibitory input rates and a, i are the amplitudes of excitatory and inhibitory postsynaptic potentials. Only for illustration, let us mention the values used in [38,39], which are a = −i = 0.2 mV and spontaneous firing activity in cortex is about 1 Hz, evoked is about 10 Hz, which gives a range for fe and fi . If we suppose, for example, that one third of synapses are inhibitory, it would induce for spontaneous activity µ = 2 mV/ms, σ 2 = 1.2 mV2 /ms, which suggests that µ and σ should be of the same order. In general, we can find a relationship based on the above reasoning; if the sizes of excitatory and inhibitory postsynaptic potentials are the same (−i = a > 0), and q > 1 is the ratio between excitation and inhibition rates (fe = qfi ), then q −1 µ , = 2 (q + 1)a σ
(36)
which gives a preliminary hint about the magnitudes of µ and σ . It suggests that the ratio between µ and σ 2 is independent of the input frequencies under the condition that while changing the frequencies, their ratio fe /fi = q is kept constant. A result which is not qualitatively different is obtained by assuming that instead of the proportional increase of the input frequencies, the increase is due to the increase of fe exclusively. In conclusion, from (36) a
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Fig. 8. Approximation of the input–output transfer function for the two-point model and its precision. The full line in the middle illustrates the approximation obtained from (35) in the case of suprathreshold stimulation of the two-point model with parameters S = 6 [mV], α = 1/3 [1/ms], αr = 1/5 [1/ms], tref = 2 [ms], (a) σ = 2 [mV/ms1/2 ], (b) σ = 4 [mV/ms1/2 ], (c) σ = 8 [mV/ms1/2 ] and it is the same input–output transfer function as in the right-hand part of Fig. 1. The crosses along this line are the outcome of the model simulations (1000 trials). The remaining√two full lines correspond the intervals (further transformed to the frequency) obtained by the crossings of the threshold by the function m2 (t) ± Var(X2 (t)). The diamond-like symbols are results of the mentioned simulation in which the average interspike interval has been increased/decreased for its two empirical standard deviation. Apparently the fit of the limits becomes good only when the first-passage-time distribution starts to be symmetrical.
crude information can be deduced about the simultaneous changes of the input parameters due to an increasing strength of stimulation. Another information about the input parameters in the single-point Ornstein-Uhlenbeck model comes immediately from the experimental data [40], where direct estimation from interspike intervals has been performed. In their paper the estimated values of the parameter µ ranges from −6.77 to 3 and σ goes up to 15. 6. Discussion As shown, the two-point model is characterized by several features that distinguishes it from its single-point counterpart, namely: the delayed reaction to stimulation onset, the lower variability of the depolarization, the lower increase of the depolarization in dependency on the stimulus intensity and the correlated voltage between the compartments which is finally reflected by the correlation structure of the interspike intervals. Let us only remind, that this correlation structure of interspike intervals cannot be produced in the single-point model, unless it is sustained by some additional features, like for example, an inhibitory feedback. The delayed reaction, which is apparent from Fig. 4 and from comparison of Eq. (2) with (28), is caused by the transfer of the input signal from the dendrite to the trigger zone which is completely neglected in the single-point model. As a consequence, the two-point model neuron is more robust against “non significant” short changes in the stimulus intensity.
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Fig. 8. (Continued )
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Fig. 9. Dependency of coefficient of variation on the firing threshold in the simulated one-point model (crosses) and two-point model (diamonds) with parameters µ = 5 [mV/ms], α = 1/3 [1/ms], αr = 1/5 [1/ms], σ = 1 [mV/ms1/2 ]. Increasing the threshold, CV tends to one for both models. The asymptotic depolarization at the trigger zone are depicted, m2 (∞) = 4.09 [mV] and m(∞) = 15 [mV].
The lower variability of the voltage fluctuation due to the randomly varying input is illustrated in Fig. 2 and it appears again in Fig. 4 and 6. It follows from (24) that the standard deviation of the steady-state depolarization in the two-point model is at least twice smaller, but in reality much lower, than the corresponding value in the single-point model. Of course, this holds under the assumption that both models are exposed to the same level of noise. If we relate the variability of the depolarization to the variability of the interspike intervals, which is legitimate, then the two-point model responds to the variability of the input by lower variability of the interspike intervals than the one-point model, again it appears to be more robust to “random” perturbations in the signal. At the first glance, this partial filtering of the input variability may seem to have the disadvantage of decreasing the coding range of the neuron. We saw in Fig. 1 that increasing σ in the single-point model results in the increase of the coding range and introduction of the second compartment partially removes this advantage. However, this decrease of the coding range is entirely compensated by the increase described by Eq. (19) and illustrated in Fig. 5. Due to it, we can see that the increase of the steady-state mean of the two-point model is at least twice lower than that in the single-point model. Consequently, the coding range is at least twice as large in the two-point model as in the single-point one. These two opposite effects (decrease of variability and decrease of the proportionality growth constant) keep the coding range at least unchanged (if not increased). From the point of view of information transfer these seem to be positive features. The last effect of the simplest spatial characterization of a neuron is the mutual dependency between the depolarizations at different compartments (Fig. 3) and consequently the serial dependency of the interspike intervals illustrated in Fig. 7. This feature, discussed in detail in [11] for the incomplete two-point model, characterizes bursting/clustering activity so often observed experimentally [41–43]. The role of bursting activity for information transfer within the nervous system is apparently enormous (only an example is LTP/LTD induction studied recently
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in [44]) and the two-point model is the simplest possible neuronal model achieving bursting for time unstructured input. Despite the obvious biological relevance of the complete two-point approach, it seems that for constant input with low variability there is only a negligible difference between the model with bi-directional connection of the compartments [13] and the model in which only one-way voltage propagation is considered [11,12] (in Fig. 6 we see no effect upon X1 caused by the reset of X2 ). One of the reasons may be the fact, that none of these models describe the shape and the effects of the action potential which is here replaced by a mere post-firing reset of the voltage. If this simplification will be removed, then the backpropagation of the action potential into the dendrite will play its well recognized role [45]. However, this modification will simultaneously remove the appreciated simplicity of the model. On the other hand, for time dependent signals considered in [13] or for description of transient states with constant stimulation, the bi-directional connection of the compartments may appear as an important factor even in this simple integrate-and-fire model. The lower depolarization at the trigger zone than at the dendrite is an inherent characteristic of the model. One of the reasons is that the model analyzed throughout this paper does not consider the soma of the neuron as its specific part, where the contributions to the trigger zone membrane depolarization are of substantial size. Whereas the input at the dendrite causes small changes of the membrane potential and thus the system is well characterized by Eq. (9), it would be natural to expect that the incoming signal located at the soma has discontinuous effect on the depolarization [46,47]. Therefore, to system (9) and (10) a third equation describing the somatic compartment may be added and this would be characterized by discontinuous trajectories, X3 (t) 2X3 (t) − X2 (t) − X1 (t) (37) + dt + adN + + idN − , dX3 (t) = − τ τr where a > 0 and i < 0 represent amplitude of excitatory and inhibitory postsynaptic potentials of synapses located at the soma and N + and N − the corresponding (homogeneous or non homogeneous in dependency in the type of input) Poisson processes driving these synapses. Similarly, to include the intrinsic noise into the model we may further modify (10) into the form X2 (t) X3 (t) − X2 (t) (38) + dt + σ2 (t)dW2 , dX2 (t) = − τ τr where σ2 (t) measures the proportion of intrinsic variability which is reflected by independent Wiener process W2 . The equation for the depolarization at the dendritic compartment would remain unchanged, except the part describing the connection with the somatic compartment, X1 (t) X3 (t) − X1 (t) + µ(t) dt + σ1 (t)dW1 . (39) + dX1 (t) = − τ τr These two modifications will change the proportion of the voltages at the different compartments and will be reconsidered in a forthcoming paper.
Acknowledgements We thank J.-P. Rospars for helpful discussions during the preparation of the paper. This work was supported by Academy of Sciences of the Czech Republic Grant No. A7011712/1997 and by GIS “Science de la Congnition” Grant CNA 10.
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