Reproducing bursting interspike interval statistics of the gustatory cortex

BioSystems 90 (2007) 442–448

Reproducing bursting interspike interval statistics of the gustatory cortex Kantaro Fujiwaraa,∗ , Hiroki Fujiwarab , Minoru Tsukadab , Kazuyuki Aiharaa,c,d a

b

Graduate School of Information Science and Technology, The University of Tokyo, 7-3-1 Hongo, Bunkyo-ku, Tokyo 113-8656, Japan Brain Science Research Center, Tamagawa University, 6-1-1 Tamagawa-gakuen, Machida, Tokyo 194-8610, Japan c Institute of Industrial Science, The University of Tokyo, 4-6-1 Komaba, Meguro-ku, Tokyo 153-8505, Japan d Aihara Complexity Modelling Project, ERATO, JST, 3-23-5 Uehara, Shibuya-ku, Tokyo 151-0064, Japan Received 22 August 2006; received in revised form 25 October 2006; accepted 26 October 2006

Abstract Cortical neurons in vivo generate highly irregular spike sequences. Recently, it was experimentally found that the local variation of interspike intervals, LV , is nearly constant for every spike sequence for the same neurons. On the contrary, the coefficient of variation, CV , varies over different spike sequences. Here, we first show that these characteristic features are also applicable in bursting spike sequences that are obtained from the rat gustatory cortex. Next, we show that the conventional leaky integrate-and-fire model does not fully account for reproducing these statistical features in data of real bursting spike sequences. We resolve this difficulty by proposing an alternative neuron model which is a reduction of the bursting neuron model involving the persistent sodium current. Our study implies that (1) the characteristic features of CV and LV are the results of the endogenous bursting and (2) the bursting behavior in the gustatory cortex is caused mainly by the persistent sodium current. © 2006 Elsevier Ireland Ltd. All rights reserved. Keywords: Bursting; Gustatory cortex; Integrate-and-fire model; Interspike intervals; Neuron

1. Introduction Neuronal bursting is a type of discharge observed in different kinds of neurons. Such bursts are thought to play important roles in electrical signaling, in neuronal synchronization, and in induction of long-term synaptic plasticity (Lisman, 1997). Several slow inward currents have been implicated in the generation of somatic bursts in cortical neurons, including voltage-sensitive calcium (Wong and Prince, 1981) and sodium currents (Azouz et al., 1996), and a calcium-activated cationic current (Kang et al., 1998). A recent study argued against the involvement of Ca2+ dependent conductances in generating fast-rhythmic burst∗

Corresponding author. Tel. +81 3 5452 6693; fax: +81 3 5452 6694. E-mail address: [email protected] (K. Fujiwara).

ing neurons, based on the observation that bursts of spikes persisted after extracellular Ca2+ was substituted with Mn2+ . Fast-rhythmic bursting neurons appeared to be replaced by slow rhythmic bursts in such a Ca2+ free condition (Mantegazza et al., 1998), which are generated by the persistent sodium current (INaP ). From this experimental observation, it is not unnatural to think that the persistent sodium current INaP plays a key role in the generation of subthreshold oscillations and burst discharge at least in some situations. From the viewpoint of theoretical studies, endogenously bursting neurons have been the subject of intensive mathematical modeling and computer simulation. However, in large number of models, calcium dependent slow variables play a key role though there exist variety kinds of ionic mechanisms to generate neuronal burst. For instance, a minimal integrate-and-

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K. Fujiwara et al. / BioSystems 90 (2007) 442–448

fire-or-burst model (Smith et al., 2000) is constructed for modeling thalamic neurons, by adding a calcium dependent inactivation T-current. On the other hand, a model of bursting generated by INaP is often very complicated. For instance, a bursting neuron model which can also reproduce fast-rhythmic bursting (so-called chattering) behavior is introduced in Wang (1999) as a two compartment model consisting 10 inward currents. To our knowledge, a good minimal bursting model which is a reduction of an INaP -based bursting model has not been proposed. A good minimal model should be a model which maintains a relation between properties of the model and experimentally measured ones, statistics of interspike intervals (ISI) for instance. One of the simplest computational neuron models, namely the leaky integrate-and-fire (LIF) model belongs to the class of single-point models, in which all the characteristics of the neuron are collapsed into a single point in space. Due to the simplicity of this model, it has long been analyzed whether it can faithfully reproduce features of real neurons (Shadlen and Newsome, 1998; Troyer and Miller, 1997; Softky, 1993; Christodoulou and Bugmann, 2001, 2000; Knoblauch and Palm, 2005). However, this model has only one variable so that there are many spiking characteristics that cannot be reproduced, and whether the LIF model is good enough is yet to be carefully examined (Feng, 2001). In examining the model plausibility, reproducing the real ISI statistics, irregularity, for example, is an important factor. The coefficient of variation CV is a very common measure which has been widely employed by many researchers (Shadlen and Newsome, 1998; Troyer and Miller, 1997; Softky, 1993; Christodoulou and Bugmann, 2001, 2000; Knoblauch and Palm, 2005). Recently, a measure of local variation of ISIs, LV , has been proposed and found to be experimentally useful (Shinomoto et al., 2003). CV and LV are defined as   n 1 1  (Ti − T¯ )2 , CV =  T¯ n − 1 i=1

n−1

1  3(Ti − Ti+1 )2 LV = , n−1 (Ti + Ti+1 )2 i=1

where Ti is the duration of the ith ISI, n the total number of ISIs, and T¯ = (1/n) ni=1 Ti is the mean ISI. CV and LV take 1 for the purely Poisson process, and 0 for perfectly periodic sequences. LV indicates the local spiking irregularity, while CV indicates the global spiking irregularity (Shinomoto et al., 2003, 2005). CV of typical cortical neurons is close to 1 (Softky, 1992). This means

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Fig. 1. CV and LV values from the bursting sequences of the gustatory cortex.

cortical neurons in vivo generates highly irregular spike sequences. We first analyze these characteristics in bursting neurons of the rat gustatory cortex. Then, we introduce a minimal INaP -based bursting neuron model instead of the LIF model to express these statistical behaviors. 2. Experimental analysis We analyze the bursting spike sequences observed in the rat gustatory cortex during a taste aversion experiment. The experimental method is explained in Appendix A. We calculated the statistical values for each bursting sequence. Fig. 1 shows the CV and LV characteristics with different average ISIs. It should be noted that LV is nearly constant under rate modulation, while CV largely varies. In addition, CV monotonically increases (statistically significant, p < 0.05 ; t test) when the average ISI increases. These results are consistent with the statistical results of non-bursting sequences. CV values from a single neuron were observed to change significantly with rate modulation, while LV is nearly the same for every spike sequence observed from the same neuron (Shinomoto et al., 2003). Additionally, CV has a property of increasing as the average ISI becoming longer in general (Softky, 1993). Fig. 1 ensures us that the CV and LV characteristics in former studies of non-bursting spike sequences are also applicable to bursting sequences in the gustatory cortex. We will examine these statistical characteristics by a simple bursting neuron model. We now summarize that the characteristics of CV and LV are as follows: (1) LV is constant compared with CV and (2) CV monotonically increases with increasing the average ISI.

3. Model analysis 3.1. LIF model We first examine whether the conventional LIF model of Eqs. (1)–(3) reproduces the characteristics of CV and

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Fig. 2. CV and LV obtained from LIF. The parameters are set as τ = 10, θ = 1.0, V0 = 0, s = 100, and Var ξ = 10.

LV . We use sinusoidal current with noise for the input (Laing and Longtin, 2003) to express the bursting behavior. Noise ξ(t) is white Gaussian with average 0, which is employed to produce various values of ISIs. dV (t) = −V (t) + aI(t), dt πt + ξ(t), I(t) = sin s if V (t) = θ, then V (t + 0) = V0 , τ

(1) (2) (3)

where V (t), τ, θ denote the membrane potential, the time constant of the membrane, and the threshold, respectively. When the membrane potential V (t) reaches the threshold θ, the neuron fires an action potential (a “spike”) and instantly resets V (t) to V0 . There are many ways to modulate ISI. For example, modulating amplitude a of the input, noise intensity, and frequency of the input. Since there is no significant difference between them according to the behaviors of CV and LV , we focus on observing the changes when modulating amplitude a of the input. Fig. 2 shows the CV and LV values obtained by modulating the amplitude a of the input in Eq. (1). When the average ISI increases, CV decreases, while LV increases in this LIF model. Additionally, LV is not constant compared with CV . These results contradict with the experimental features.

Fig. 3. Average ISIs and average inter-burst intervals obtained from the experimental data (dots) and the LIF model (dotted line). For the experimental data, the inter-burst interval is variable for modulating ISI, which is different from the LIF model.

LIF model. In the LIF model, inter-burst interval does not change since it is difficult to fire while the sinusoidal input is small. Let us consider the statistical values in very simple bursting sequences, where the duration of bursting is T, the inter-burst interval with no spikes is also T, and a neuron periodically switches the bursting and resting phases (see Fig. 4). If the number of spikes per burst is n + 1, the intra-burst interval would be T/n. In this case, the average ISI is modulated by the spike number per each burst. CV,n = (n − 2)/ √If there are n spikes in each burst, (2 n − 1), and L√ 1)2 )/(n(n +√1)2 ), CV, V,n = (3(n − √ n + 1 − CV,n = (n n − (n − 1) n + 1)/2 n(n + 1) > 0 for ∀n ∈ N. Therefore, if the spike number per burst increases from n to n + 1, CV increases. On the other hand, LV,n+1 − LV,n = 3(n3 (n + 1) − (n − 1)2 (n + 2)2 )/(n(n + 1)2 (n + 2)2 ) < 0 for ∀n ∈ N, n > 3. Therefore, LV decreases if the spike number per burst increases over 3. The increase of the spike number means the average ISI shortens, thus if ISI shortens, CV increases while LV decreases. If the average ISI shortens, the intra-burst interval becomes shorter. Since CV mainly measures a difference between the average ISI and each intra-burst interval in this case, increase of those differences results in increase of CV . On the other hand, LV is almost constant since

3.2. What makes CV and LV oppositely behave? What is the difference between the LIF model and the real neurons? Fig. 3 shows the relation between ISIs and inter-burst intervals obtained from the experimental data and the LIF model. For the experimental data, the inter-burst interval is variable for modulating ISI, which is different from the

Fig. 4. A simple bursting sequence with n spikes per each burst.

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it measures a difference between the intra-burst interval and the inter-burst interval. Due to these facts, CV and LV show opposite properties for increasing the ISI. However, what mechanism causes CV to decrease when ISI shortens in bursting sequences of real neurons? In the case of the simple example above, the inter-burst interval was kept constant. Variable inter-burst intervals make the difference between the average ISI and each intra-burst interval increase when the average ISI increases. Thus, CV increases as the average ISI shortens. Irreproducibility of the CV property in the LIF model is also due to this reason. In the LIF model, firing is completely affected by the input, so that if there is no input, it does not fire at all. Therefore, when the inter-burst interval is kept constant, the result is inconsistent with the property of the CV in the experimental data. The property of the CV implies that the modulation of inter-burst intervals is necessary for ISI modulation. Constant LV also implies the same condition. Large LV fluctuation does not occur when the inter-burst intervals vary. Variable inter-burst intervals in experimental data imply that the bursting neuron model needs the mechanism to fire not only by the input. 3.3. Alternative neuron model We introduce the following alternative bursting neuron model for reproducing experimental statistics: dV (t) = −V (t) + η(t) + aI(t), dt η(t) = −λt, τ

if V (t) = θ,

then V (t + 0) = V0 ,

(4) (5) η(t + 0) = η0 ,

(6) where V (t), τ, θ denote the membrane potential, the time constant of the membrane, and the threshold, respectively. This is the model with variable η(t) added to the LIF model. The dynamics of this model is also very simple. The LIF model can be interpreted as a motion of a particle in a potential. If it reaches the threshold (=fire), it resets to a resting potential. In the new model, η(t) takes a positive value η0 after firing, so that a particle gets easily to climb the potential for a while. This voltage non-resetting mechanism causes positive ISI correlation. Since η(t) decays with a constant rate λ, it prevents too strong correlation. It is η that makes inter-burst intervals variable, thus preventing CV from the wrong behavior. Fig. 5 shows the CV and LV characteristics obtained from the new model. These characteristics can be obtained in a wide range of the parameter values. If we modulate λ and η0 while

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Fig. 5. CV and LV values obtained from the new model. Standard deviations of CV and LV are 0.130 and 0.041, respectively. Therefore, CV varies largely compared with LV . λ = 0.001, η0 = 1.0, and other parameters are the same with the LIF model in Fig. 2.

the other parameter values are the same with Figs. 2 and 5, these characteristics are realized in the range of η0 /λ > 142.85. Since η is positive for η0 /λ ms after firing, η0 /λ indicates a persistency of η. Namely, the persistency over 142.85 ms is necessary for the CV and LV behaviors. This model produces a positive correlation coefficient even with uncorrelated inputs. This means that the model is able to generate the spike correlation by itself. Since the new model resets η to a positive value after firing, it has a tendency to fire easily after firing, which produces spike correlation. η(t) can be interpreted as the partial resetting of the voltage, which is observed in many realistic neurons (Segundo et al., 1967). Though there exists a neuron model which employs a partial reset mechanism to express irregular firing (Bugmann et al., 1997), our proposed model is much simpler. Moreover, our proposed model is a generalization of the conventional LIF model. Setting parameters η0 = 0 and λ = 0, we can neglect the η dynamics. In this case, this model is equivalent to the LIF model. We can interpret η(t) as internal dynamics rather than external input since it depends on the spike generation, which is not natural for the external phenomenon. 4. Model interpretation In our model, various kinds of bursting behaviors are easily obtained by adjusting parameters. The decaying rate λ and the reset value η0 determine the firing patterns, namely bursting or regular firing. Even high frequency bursting as in our experiment is reproducible by our model, which is generally difficult for bursting neuron models.

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A fast-rhythmic bursting neuron is well known as the chattering neuron, which is a promising candidate for the pacemaker of coherent gamma-band (25–70 Hz) cortical oscillation. It is still in controversy how the neuron generates such bursts. In Wang’s model (Wang, 1999), the bursting mechanism involves a “ping-pong” interplay between soma-to-dendrite back propagation of action potentials and afterdepolarization generated by a persistent dendritic sodium current and a somatic sodium window current. The model has two compartments, representing the dendrite and the soma, respectively. The dendritic compartment has a persistent sodium current INaP and a slowly inactivating potassium current IKS predominantly. Repetitive bursting is primarily generated by the interplay between action potentials in the somatic compartment and slow active currents in the dendritic compartment. In our model, resetting η to a positive value η0 after firing helps depolarizing after firing, and η decaying by rate λ helps ceasing bursting, resulting in firing frequency adaptation. This dynamics of decaying η has strong similarity with the dendrite-to-soma current of Wang’s model, which involves persistent sodium current INaP and slowly inactivating potassium current IKS (see Fig. 6). There exist other kinds of fast-rhythmic bursting neuron models that focus on the other ionic dynamics, the

Fig. 7. Average spike number per burst with modulating inter-burst intervals.

Ca2+ dependent cationic current, for example Aoyagi et al. (2003). Ca2+ is kept low in the inter-burst intervals, irrelevant to the applied current intensity. On the other hand, the effect of INaP remains activated not only during each burst but also in the intervals between the bursts. These differences between voltage dependent INaP and Ca2+ dependent cationic current underlie the qualitative differences of the bursting behaviors between the INaP -based model and the Ca2+ dependent cationic current-based ones. In the former, the number of spikes per burst necessarily increases with an increase in the applied current intensity. In the latter, the number remains unchanged in a wide range of bursting frequency. Fig. 7 shows the number of spikes per burst for modulating inter-burst intervals in both the experiment and our model. The consistency of the features of the experimental data and our model indicates that our model is a reduction of the INaP -based model in some aspects. 5. Discussion

Fig. 6. Similarity of the dendrite-to-soma current in INaP -based model (above) and η in our model.

Spike production can be considered as a two-step process. First, synaptic inputs are integrated by an extensive and complex dendritic tree resulting in a total synaptic current at a trigger zone. Second, the cell emits spikes in response to this total synaptic current. A numerous number of single neuron models that can reproduce some aspects of spiking statistics of biological neurons, such as the probability density of interspike intervals (Lindner, 2004) have been produced, and most of them have largely focused on the latter. We have shown that the statistical features of non-bursting ISI sequences are also applicable to the bursting ISI sequences. However, the LIF model does not account for the experimental data, with respect to reproducing the ISI statistics, CV and LV . On the other hand, our proposed neuron model which considers the endogenous dynamics, which is not perfectly dependent only on voltage dynamics is able to reproduce them. This difference of reproducibility

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has been ascribed to its spike generation mechanism. The CV and LV behaviors of our model suggest that the endogenous mechanism plays a significant role in producing output temporal correlation in bursting neurons, rather than the synaptic inputs does. This indicates that it is possible to regard a bursting neuron as a unit to perform temporal operations dependent on the internal dynamics instead of stochastic operations through the presynaptic inputs at each instance. This is consistent with many biological data that exhibit an anomalously large value of positive correlation coefficient (Nelson and Maciver, 1999), temporal correlation without inputs (Ratnam and Nelson, 2000), and the complex temporal operations in a single neuron level (Matsumoto et al., 1987). It is an interesting future problem to consider whether these assumptions in bursting neurons are applicable to non-bursting, regular neurons. Additionally, our study suggests a possibility that a bursting behavior of neurons in the gustatory cortex is mainly caused by INaP . The mechanism of the gustatory bursting behavior has not yet clarified physiologically, but the statistics of bursting ISI sequences implies that the main source of the burst is INaP , which is often observed in the other cortical areas. According to Section 3.3, the persistency of INaP is evaluated as 142.85 ms, which is also plausible in other cortical areas. The persistency may be a consequence of a background activity. Although we analyzed the bursting spike sequences observed in the rat gustatory cortex during a taste aversion experiment, the behaviors of CV and LV were similar to the controlled data. This means that INaP also exists in the controlled experiment and invariably in the rat gustatory cortex. 142.85 ms of the persistency can be interpreted as the consequence of the brain wave activity with the θwave (4–8 Hz) or the α-wave (8–14 Hz). It is also a future problem to confirm these assumptions experimentally. Consequently, our proposed model is a generalized form of the LIF model, and the biological plausibility may be high with holding the low implementation cost. Comparing with other neuron models, various patterns of firing are reproducible by the model, e.g., the ability of expressing tonic bursting and the chattering behavior.

Appendix A. Taste aversion experiment

Acknowledgments

References

This research is partially supported by Grant-in-Aid for Scientific Research on Priority Areas 17022012 from MEXT of Japan and by the Superrobust Computation Project in 21st Century COE Program on Information Science and Technology Strategic Core from the Ministry of Education, Culture, Sports, Science, and Technology, the Japanese Government.

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Subjects were male Long-Evans rats weighing between 270 and 320 g. They were neither juvenile nor aged animals (they were about 10 weeks old). They were caged in pairs and housed at a constant temperature (22 ± 2 ◦ C) in an animal-keeping room maintained under a 14-h light:10-h dark cycle. They received ad lib food and water until surgery. After surgery, they were caged individually. During the experiments after training, access to water was limited to an evening drinking period of 20 min. Anesthetized rats [ketamine and xylazine (100 and 7 mg/kg, i.p., respectively)] were implanted unilaterally or bilaterally with electrode bundles in the gustatory cortex (GC: anterior, 1.0–1.5 mm; mediolateral, 5.0 mm; dorsoventral, approximately 5.5 mm from skull). Test cages were made of transparent acrylic boxes (D: 240 mm, W: 300 mm, H: 400 mm) without roof. On the front wall, there were two holes (20 mm in diameter) located 50 mm above the floor and 90 mm apart. These holes allowed graduated drinking tubes (glass Richter tubes) to be inserted into the cage, providing access to a fluid (always presented at room temperature). A headstage with 12 independently mobile microdrives containing tetrodes (280–320 k ) was implanted on the skull of each rat before behavioral training. Neuronal signals were fed through amplifiers (gain, 10,000; filters, 0.6–9 kHz) to computers running customized data acquisition software that captured spikes exceeding a preset voltage threshold at 32 kHz per channel and stored the digitized data with time stamps. Bursting sequences are extracted by two steps. First, we randomly selected the sequences which contain hundred ISIs from each neuron and each task. Next, we regarded the spike sequence as the bursting sequence if the spike sequence contains more than five pieces of consecutive short (under 5 ms) ISIs. Five hundred and twenty eight bursting sequences have been obtained in this method, including the sequences bursting with very high frequency. Inter-burst intervals are defined as the intervals between them.

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