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A single spike deteriorates synaptic conductance estimation
Accepted Manuscript Title: A single spike deteriorates synaptic conductance estimation Author: Ryota Kobayashi Hiroshi Nishimaru Hisao Nishijo Petr Lansky PII: DOI: Reference:
S0303-2647(17)30069-2 http://dx.doi.org/doi:10.1016/j.biosystems.2017.07.007 BIO 3758
To appear in:
BioSystems
Received date: Revised date: Accepted date:
28-2-2017 19-7-2017 20-7-2017
Please cite this article as: Ryota Kobayashi, Hiroshi Nishimaru, Hisao Nishijo, Petr Lansky, A single spike deteriorates synaptic conductance estimation, (2017), http://dx.doi.org/10.1016/j.biosystems.2017.07.007 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
A single spike deteriorates synaptic conductance estimation
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Ryota Kobayashia,b , Hiroshi Nishimaruc , Hisao Nishijoc , Petr Lanskyd a
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Principles of Informatics Research Division, National Institute of Informatics, 2-1-2 Hitotsubashi, Chiyoda-ku, Tokyo, Japan b Department of Informatics, Graduate University for Advanced Studies (Sokendai), 2-1-2 Hitotsubashi, Chiyoda-ku, Tokyo, Japan c System Emotional Science, Graduate School of Medicine and Pharmaceutical Sciences, University of Toyama, Sugitani 2630, Toyama 930-0194, Japan d Institute of Physiology, The Czech Academy of Sciences, 142 20 Prague 4, Czech Republic
Abstract
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We investigated the estimation accuracy of synaptic conductances by analyzing simulated voltage traces generated by a Hodgkin−Huxley type model. We show that even a single spike substantially deteriorates the estimation. We also demonstrate that two approaches, namely, negative current injection and spike removal, can ameliorate this deterioration.
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Keywords: synaptic conductance estimation, single neuron models, Ornstein−Uhlenbeck process 1. Introduction
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It is essential to estimate synaptic conductances from experimental data for understanding computations in neural circuits. A method to determine synaptic conductances was developed on the basis of fitting voltage clamp data (Borg-Graham et al., 1998; Monier et al., 2008; Puggioni et al., 2017). Although this method can provide insight into the underlying synaptic dynamics, it requires a sophisticated experimental technique, that is, a large number of voltage clamp recordings at multiple holding potentials. To overcome the technical difficulty, several other methods (B´edard et al., 2012; Paninski et al., 2012; Berg and Ditlevsen, 2013; Lankarany et al., 2013, 2016; Kobayashi et al., 2016) have been proposed for estimating the synaptic
Preprint submitted to BioSystems
July 19, 2017
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conductances from single voltage traces, which are relatively easy to obtain experimentally. These estimation methods are based on the leaky integrator model described by a linear differential equation. This model is simple and widely used in computational neuroscience (Gerstner and Kistler, 2002). An alternative to this neuron model is the Hodgkin−Huxley-type models, which are more realistic in the sense that they incorporate voltage-gated ion channels of a neuron. Even though it has been shown that the leaky integrator model can approximate the Hodgkin−Huxley models (Abbott and Kepler, 1990; Kistler et al., 1997; Jolivet et al., 2004; Kobayashi and Kitano, 2016) and reproduce experimental data accurately (Bugmann et al., 1997; Jolivet et al., 2006; Kobayashi et al., 2009), it is unclear how the ion channels affect the conductance estimation. In this study, we investigated the estimation accuracy of synaptic conductances by using simulated voltage traces generated by a Hodgkin−Huxley type model. We show that even a single spike deteriorates substantially the estimation, and we propose two approaches to improve the estimation in the firing regime. 2. Methods
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2.1. Estimating synaptic conductance from a voltage trace First, we summarize the procedure for estimating synaptic conductance from a voltage trace (see Berg and Ditlevsen (2013), for details). Let us consider an in-vivo like situation in which a neuron receives excitatory and inhibitory inputs. Below the spike threshold, the membrane voltage of a neuron V (t) can be described by a linear differential equation (Gerstner and Kistler, 2002; Kobayashi and Kitano, 2016): dV = −gL (V − VL ) − gE (t)(V − VE ) − gI (t)(V − VI ) + Iinj , dt
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(1)
where Cm is the membrane capacitance, gL is the leak conductance, and VL , VE , and VI are the leak, excitatory, and inhibitory reversal potentials, respectively. In addition, an external current Iinj is injected in some experiments to prevent the recorded neuron from firing (Kobayashi et al., 2016). Excitatory and inhibitory synaptic conductances (gE (t), gI (t)) are input signals to the recorded neuron, and their estimation is our focus here. We suppose that the mean of the synaptic conductances are stationary within an observation window [0, w], namely, gE (t) = g¯E + ηE (t) and gI (t) = 2
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g¯I + ηI (t), where g¯E and g¯I are the mean conductances, and ηE (t) and ηI (t) are unspecified random fluctuations. The window size was set to w = 0.4 s. The Ornstein−Uhlenbeck model approximates Eq. (1) by replacing the fluctuation terms with Gaussian white noise, ( ) dV Cm = −¯ gtot V − V¯tot − ηE (t)(V − VE ) − ηI (t)(V − VI ) dt ( ) ≈ −¯ gtot V − V¯tot + σtot ξ(t), (2)
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where ξ(t) is the Gaussian white noise of zero mean and unit standard deviation (SD). The total conductance g¯tot and the effective reversal potential V¯tot are defined as g¯tot = gL + g¯E + g¯I , V¯tot = (gL VL + g¯E VE + g¯I VI + Iinj )/¯ gtot .
(3) (4)
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Supposing that the voltage is in equilibrium ( dV ≈ 0), the effective reverdt sal potential can be estimated from the sample mean in the window: 1∑ Vˆtot = Vj , n j=1 n
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(5)
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where Vˆtot denotes the estimator, Vj = V (j∆t) is the recorded voltage at time j∆t, ∆t = 0.1 ms is the sampling interval, and n is the number of samples within the window. In this study, we take whole the observation period for the window (n = 4, 000). The total conductance can be estimated by fitting the auto-correlation function to the exponential function ( ) gˆtot k∆t ρˆk ∼ ρ0 exp − (6) , Cm
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where ρˆk = ρ(k∆t) is the auto-correlation function of the voltage with lag k∆t, and gˆtot denotes the estimator for the total conductance. Following Berg and Ditlevsen (2013), the auto-correlation function for small k (0 ≤ k ≤ 40 corresponding to 0−4 ms) was used for the conductance estimation. We confirmed that the choice of the fitting interval does not influence the results qualitatively (Data not shown). The auto-correlation function is calculated by the sum of the sample correlation function and a correction term: )( ) ∑n−k ( ˆ ˆ V − V V − V j tot j+k tot j=1 2k + . (7) ρˆk = ( ) 2 ∑n n−1 ˆ V − V j tot j=1 3
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We estimate the excitatory and inhibitory synaptic conductances after determining the total conductance and the effective reversal potential. The estimators follow from Eqs. (3) and (4): (8)
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gˆtot (Vˆtot − VI ) − gL (VL − VI ) − Iinj , VE − VI gˆI = gˆtot − gˆE − gL .
gˆE =
(9)
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In this paper, the parameters of the estimation model are Cm = 1.0 nF, gL = 0.1 µS, VL = −70 mV, VE = 0 mV, and VI = −75 mV. These parameters are assumed to be available, because it is possible to measure them by different experiments.
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2.2. Simulation model The leaky integrator model and a Hodgkin−Huxley type model were used to investigate the estimation performance of the conductance estimation method (see Section 2.1).
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2.2.1. Leaky integrator model The voltage V (t) of the leaky integrator model neuron is described by Eq. (1). The membrane parameters were set to Cm = 1.0 nF, gL = 0.1 µS, VL = −70 mV, VE = 0 mV, and VI = −75 mV. The injected current Iinj was zero, unless otherwise stated. The synaptic conductances were described by the point-conductance model (Destexhe et al., 2001), √ dgE 2 gE − g¯E =− + σE ξE (t), (10) dt τE τE √ dgI 2 gI − g¯I =− + σI ξI (t), (11) dt τI τI
where g¯E(I) and σE(I) are the mean and SD of the synaptic conductance, τE(I) = 0.5 (1.0) ms is the synaptic time constant, and ξE(I) (t) is the independent Gaussian white noise with zero mean and unit SD. The SD of the excitatory conductance was assumed to be proportional to the corresponding mean, σE = 0.2¯ gE (Miura et al., 2007). The mean and SD of the inhibitory conductance were g¯I = 50 nS and σI = 10 nS. Eqs. (1), (10), and (11) were solved numerically using the forward Euler integration method with a time step of 0.01 ms. 4
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dV dt
= −gL (V − VL ) − INa − IK
−gE (t)(V − VE ) − gI (t)(V − VI ) + Iinj .
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2.2.2. Hodgkin−Huxley type model We used a Hodgkin−Huxley type model of a cortical neuron proposed by Pospischil et al. (2008), which is biophysically realistic compared with the leaky integrator model (Eq. 1). The voltage V of the model neuron receiving excitatory and inhibitory inputs was described by
(12)
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The passive parameters (Cm , gL , VL , VE , and VI ) were the same as in the leaky integrator model (Section 2.2.1). The Na+ current INa was described by (13)
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INa = gNa m3 h(V − ENa ),
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dm = αm (V )(1 − m) − βm (V )m, dt −0.32(V + 45) αm (V ) = −(V +45)/4 , e −1 0.28(V + 18) βm (V ) = (V +18)/5 , e −1 dh = αh (V )(1 − h) − βh (V )h, dt αh (V ) = 0.128e−(V +41)/18 , 4 βh (V ) = , −(V 1 + e +18)/5
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where gNa = 50 µS, and ENa = 50 mV. The delayed-rectifier K+ current IK was described by IK = gK n4 (V − EK )
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dn = αn (V )(1 − n) − βn (V )n, dt −0.032(V + 43) , αn (V ) = −(V +43)/5 e −1 βn (V ) = 0.5e−(V +48)/40 , 5
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where gK = 5 µS, and EK = −90 mV. The synaptic conductances (gE (t), gI (t)) were described by the point conductance model (Eqs. 10 and 11), and the parameters were the same as in Section 2.2.1. This model was solved numerically using the forward Euler integration method with a time step of 0.01 ms. Fig. 1 shows the voltage traces generated by the two model neurons receiving excitatory and inhibitory inputs. Two levels of excitatory inputs are injected into these model neurons, and the voltage of both model neurons increases with the excitatory input. The Hodgkin−Huxley model occasionally generates spikes for the stronger input (Fig. 1b). In contrast, the leaky integrator model does not generate spikes, because it does not incorporate the spike generation mechanism.
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Figure 1: Simulated voltage traces for the weak (black) and strong (red) excitatory input. (a) Leaky integrator model. (b) A Hodgkin−Huxley type model. The model generates a spike at t = 0.15 s for the strong input. The voltage traces are shown for 0.4 s, which is the observation window used in our analysis.
2.3. Estimation errors Estimation errors of the excitatory and inhibitory conductances were evaluated by the mean relative error (MRE), M 1 ∑ gˆk − g¯ , MRE = M k=1 g¯
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and the mean absolute relative error (MARE),
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M 1 ∑ gˆk − g¯ MARE = , M k=1 g¯
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where gˆk is an estimate of the mean conductance in the k-th trial and g¯ is the true mean. MRE measures the estimation bias, whereas MARE measures the sum of the bias and the variability of an estimator. The errors were calculated from M = 1, 000 trials, unless otherwise stated.
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3. Results
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3.1. A single spike deteriorates the synaptic conductance estimation Simulated data were generated by injecting excitatory and inhibitory conductance inputs into the leaky integrator model (Section 2.2.1) and into a Hodgkin−Huxley type model (Section 2.2.2). Fig. 2 shows the effect of excitatory input on the conductance estimation errors evaluated by the mean relative error (MRE) and the mean absolute relative error (MARE). The errors for the leaky integrator model are almost constant over the whole range of excitation. The relative errors are negative, which indicates that the estimates are smaller than the true values. The two errors (MRE and MARE) are -10 % and 20 % for excitatory conductance, and -20% and 50 % for inhibitory conductance, respectively. For weak inputs (¯ gE ≤ 30 nS), the errors for the Hodgkin−Huxley model are also almost constant. As the excitatory input increases, the relative error decreases slightly and then increases sharply around g¯E ≈ 40 nS. The relative errors for the Hodgkin−Huxley model are positive for strong inputs, which indicates that the estimates are larger than the true values. The errors for strong inputs (¯ gE > 40 nS) are not shown in Fig. 2, because they are larger than 200 %. For weak inputs (¯ gE < 40 nS), the errors for the Hodgkin−Huxley model are comparable to those for the leaky integrator model. In contrast, the errors for the Hodgkin−Huxley model are much larger than those for the leaky integrator model for strong inputs (¯ gE > 40 nS). As shown in Fig. 1, the Hodgkin−Huxley model occasionally discharges spikes for the strong inputs. This result suggests that the spikes lead to a large increase in the errors of conductance estimation.
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Figure 2: Effect of excitatory input on the synaptic conductance estimation. (a, b) Dependence of the errors (a: MRE, b: MARE) for the excitatory conductance gE on the input conductance g¯E . (c, d) Dependence of the errors (c: MRE, d: MARE) for the inhibitory conductance gI on the input conductance g¯E . The errors for the leaky integrator model (LI: black) are compared with those of the Hodgkin−Huxley model (HH: red dashed).
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In the following, we focus on the estimation errors for the Hodgkin−Huxley model, because the effect of excitatory inputs on the estimation errors is modest for the leaky integrator model (Fig. 2). Fig. 3 shows the effect of spikes on the estimation errors. The errors increase immediately when the model starts to generate spikes. When the excitatory input is intermediate (40 < g¯E ≤ 45), the model generates spikes in only some trials. The estimation errors of no spike trials are much smaller than those of all the trials. Table 1 summarizes the errors for a sub-threshold input (¯ gE = 35 nS) and the errors for an intermediate input (¯ gE = 45 nS) where the average number of spikes was 1.0. Although the duration of a spike is very short (less than 1 % of the observation period), even a single spike markedly increases the errors. It is known that the Hodgkin−Huxley model exhibits nonlinear behavior even in the sub-threshold regime (Jolivet and Gerstner, 2004). The mean absolute errors of ”no spike” trials slightly increase with the excitatory input (Fig. 3 b, d), which is considered to be due to the sub-threshold nonlinearity. The result shows that the effect of this sub-threshold nonlinearity on the estimation accuracy is modest compared to the effect of a single spike (Fig. 3 and table 1). The reason may be that Equation (1) is rather good approximation of the Hodgkin-Huxley model in the sub-threshold regime but not during the spike itself. The effect of a single spike on the auto-correlation function of the voltage is shown in Fig. 4. The estimates of the total conductance from the voltage trace with a spike was 990 nS, whereas that with no spike was 120 nS. Thus, the synaptic conductance is largely over-estimated in the trials containing a spike. This result suggests that even a single spike significantly deteriorates the conductance estimation. Here, we considered only the increasing excitatory input. However, the result does not change quantitatively for the balanced input (Petersen et al., 2014; Kobayashi et al., 2015) where the excitatory and inhibitory input increase simultaneously (data not shown).
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Table 1: Estimation errors of synaptic conductance for the Hodgkin−Huxley model. Simulated data were generated from two levels of excitatory input (¯ gE = 35 and 45 nS). For the stronger input, the errors were calculated from no spike trials (¯ gE = 45 nS, no spikes) and from all the trials (¯ gE = 45 nS).
MRE (Inh) −39 −120 1, 100
MARE (Exc) MARE (Inh) 23 56 43 120 500 1, 200
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MRE (Exc) g¯E = 35 nS −16 g¯E = 45 nS, No spikes −43 g¯E = 45 nS 500
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Figure 3: Effect of spikes on the synaptic conductance estimation in the Hodgkin−Huxley model. (a, b) Dependence of the errors (a: MRE, b: MARE) for the excitatory conductance gE on the input conductance g¯E . (c, d) Dependence of the errors (c: MRE, d: MARE) for the inhibitory conductance gI on the input conductance g¯E . The mean errors of the trials with no spikes (gray) are compared with those of all the trials (red dashed). The arrows indicate the minimal input conductance required for generating spikes. (e, f) Dependence of the average number of spikes Nsp on the input conductance g¯E .
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Figure 4: Effect of a single spike on the auto-correlation function. (a) Simulated voltage trace with a spike (red) and in absence of spikes (blue). (b) Corresponding auto-correlation functions (ACF) using the same colors as in (a). The exponential fits to the ACF are shown in gray. The excitatory input was g¯E = 45 nS for both voltage traces.
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3.2. Approaches to estimate synaptic conductance in the firing regime We have shown that even a single spike causes significant deterioration in the synaptic conductance estimation. Here, we consider two approaches to reduce the deterioration effect. The first approach is to inject a negative current into the neuron to prevent spike discharges, which has been traditionally used in electrophysiological studies (Kobayashi et al., 2016). We confirm that the negative current injection markedly reduces the estimation errors of the synaptic conductances (Fig. 5, Table 2). The second approach is to remove spike waveforms from the voltage trace, which was suggested by Lansky et al. (2006) and Berg and Ditlevsen (2013). The synaptic conductances were estimated from sub-threshold parts of the voltage trace, namely V (t) < −40 mV. We again confirm that the spike removal also reduces the estimation errors (Table 2). The ameliorating effect of negative current injection is larger than the effect of spike removal.
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Table 2: Comparison of the approaches to recover from the deterioration in the conductance estimation caused by spikes. Estimation errors are compared among the three methods: estimation from the whole voltage trace (Original), estimation from the whole voltage trace with negative current injection Iinj = −1.0 nA (Current injection), and estimation from the sub-threshold voltage trace (Spike removal).
MRE (Inh) 1, 100 −26 −85
MARE (Exc) 520 15 31
MARE (Inh) 1, 200 57 85
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MRE (Exc) Original 500 Current injection −7 Spike removal −30
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Figure 5: Negative current injection can recover from the deterioration in the conductance estimation caused by spikes. (a, b) Dependence of the estimation errors (a: MRE, b: MARE) on the injected current Iinj in Eq. (12). Cyan and dashed magenta represent the errors of excitatory and inhibitory conductance, respectively. (c) Dependence of the average number of spikes on the injected current. The mean excitatory conductance was g¯E = 45 nS. The arrows indicate the minimal injected current required for generating spikes. Note that the negative current (Iinj < 0) was injected to prevent spike generation.
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4. Conclusions
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We have shown that a single spike deteriorates the estimation of synaptic conductance. The effect of a spike is much larger than the effect of subthreshold nonlinearity in the Hodgkin−Huxley model. We have also proposed two approaches to recover from this deterioration caused by spikes: negative current injection and spike removal. It is confirmed that these approaches dramatically reduced the estimation error. As a future work, we are planning to investigate how to remove spikes from a voltage trace for an accurate conductance estimation.
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Acknowledgements We thank two anonymous reviewers for their valuable comments. This work was supported by ACT-I, Japan Science and Technology Agency, JSPS KAKENHI Grant Number 25870915 and 15K06695, the Okawa Foundation for Information and Telecommunications and the open collaborative research and MOU grant at National Institute of Informatics in Japan. Supported by the Institute of Physiology RVO:67985823 and the Czech Science Foundation project 15-08066S.
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S0303-2647(17)30069-2 http://dx.doi.org/doi:10.1016/j.biosystems.2017.07.007 BIO 3758
To appear in:
BioSystems
Received date: Revised date: Accepted date:
28-2-2017 19-7-2017 20-7-2017
Please cite this article as: Ryota Kobayashi, Hiroshi Nishimaru, Hisao Nishijo, Petr Lansky, A single spike deteriorates synaptic conductance estimation, (2017), http://dx.doi.org/10.1016/j.biosystems.2017.07.007 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
A single spike deteriorates synaptic conductance estimation
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Ryota Kobayashia,b , Hiroshi Nishimaruc , Hisao Nishijoc , Petr Lanskyd a
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Principles of Informatics Research Division, National Institute of Informatics, 2-1-2 Hitotsubashi, Chiyoda-ku, Tokyo, Japan b Department of Informatics, Graduate University for Advanced Studies (Sokendai), 2-1-2 Hitotsubashi, Chiyoda-ku, Tokyo, Japan c System Emotional Science, Graduate School of Medicine and Pharmaceutical Sciences, University of Toyama, Sugitani 2630, Toyama 930-0194, Japan d Institute of Physiology, The Czech Academy of Sciences, 142 20 Prague 4, Czech Republic
Abstract
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We investigated the estimation accuracy of synaptic conductances by analyzing simulated voltage traces generated by a Hodgkin−Huxley type model. We show that even a single spike substantially deteriorates the estimation. We also demonstrate that two approaches, namely, negative current injection and spike removal, can ameliorate this deterioration.
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Keywords: synaptic conductance estimation, single neuron models, Ornstein−Uhlenbeck process 1. Introduction
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It is essential to estimate synaptic conductances from experimental data for understanding computations in neural circuits. A method to determine synaptic conductances was developed on the basis of fitting voltage clamp data (Borg-Graham et al., 1998; Monier et al., 2008; Puggioni et al., 2017). Although this method can provide insight into the underlying synaptic dynamics, it requires a sophisticated experimental technique, that is, a large number of voltage clamp recordings at multiple holding potentials. To overcome the technical difficulty, several other methods (B´edard et al., 2012; Paninski et al., 2012; Berg and Ditlevsen, 2013; Lankarany et al., 2013, 2016; Kobayashi et al., 2016) have been proposed for estimating the synaptic
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July 19, 2017
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conductances from single voltage traces, which are relatively easy to obtain experimentally. These estimation methods are based on the leaky integrator model described by a linear differential equation. This model is simple and widely used in computational neuroscience (Gerstner and Kistler, 2002). An alternative to this neuron model is the Hodgkin−Huxley-type models, which are more realistic in the sense that they incorporate voltage-gated ion channels of a neuron. Even though it has been shown that the leaky integrator model can approximate the Hodgkin−Huxley models (Abbott and Kepler, 1990; Kistler et al., 1997; Jolivet et al., 2004; Kobayashi and Kitano, 2016) and reproduce experimental data accurately (Bugmann et al., 1997; Jolivet et al., 2006; Kobayashi et al., 2009), it is unclear how the ion channels affect the conductance estimation. In this study, we investigated the estimation accuracy of synaptic conductances by using simulated voltage traces generated by a Hodgkin−Huxley type model. We show that even a single spike deteriorates substantially the estimation, and we propose two approaches to improve the estimation in the firing regime. 2. Methods
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2.1. Estimating synaptic conductance from a voltage trace First, we summarize the procedure for estimating synaptic conductance from a voltage trace (see Berg and Ditlevsen (2013), for details). Let us consider an in-vivo like situation in which a neuron receives excitatory and inhibitory inputs. Below the spike threshold, the membrane voltage of a neuron V (t) can be described by a linear differential equation (Gerstner and Kistler, 2002; Kobayashi and Kitano, 2016): dV = −gL (V − VL ) − gE (t)(V − VE ) − gI (t)(V − VI ) + Iinj , dt
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(1)
where Cm is the membrane capacitance, gL is the leak conductance, and VL , VE , and VI are the leak, excitatory, and inhibitory reversal potentials, respectively. In addition, an external current Iinj is injected in some experiments to prevent the recorded neuron from firing (Kobayashi et al., 2016). Excitatory and inhibitory synaptic conductances (gE (t), gI (t)) are input signals to the recorded neuron, and their estimation is our focus here. We suppose that the mean of the synaptic conductances are stationary within an observation window [0, w], namely, gE (t) = g¯E + ηE (t) and gI (t) = 2
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g¯I + ηI (t), where g¯E and g¯I are the mean conductances, and ηE (t) and ηI (t) are unspecified random fluctuations. The window size was set to w = 0.4 s. The Ornstein−Uhlenbeck model approximates Eq. (1) by replacing the fluctuation terms with Gaussian white noise, ( ) dV Cm = −¯ gtot V − V¯tot − ηE (t)(V − VE ) − ηI (t)(V − VI ) dt ( ) ≈ −¯ gtot V − V¯tot + σtot ξ(t), (2)
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where ξ(t) is the Gaussian white noise of zero mean and unit standard deviation (SD). The total conductance g¯tot and the effective reversal potential V¯tot are defined as g¯tot = gL + g¯E + g¯I , V¯tot = (gL VL + g¯E VE + g¯I VI + Iinj )/¯ gtot .
(3) (4)
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Supposing that the voltage is in equilibrium ( dV ≈ 0), the effective reverdt sal potential can be estimated from the sample mean in the window: 1∑ Vˆtot = Vj , n j=1 n
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(5)
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where Vˆtot denotes the estimator, Vj = V (j∆t) is the recorded voltage at time j∆t, ∆t = 0.1 ms is the sampling interval, and n is the number of samples within the window. In this study, we take whole the observation period for the window (n = 4, 000). The total conductance can be estimated by fitting the auto-correlation function to the exponential function ( ) gˆtot k∆t ρˆk ∼ ρ0 exp − (6) , Cm
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where ρˆk = ρ(k∆t) is the auto-correlation function of the voltage with lag k∆t, and gˆtot denotes the estimator for the total conductance. Following Berg and Ditlevsen (2013), the auto-correlation function for small k (0 ≤ k ≤ 40 corresponding to 0−4 ms) was used for the conductance estimation. We confirmed that the choice of the fitting interval does not influence the results qualitatively (Data not shown). The auto-correlation function is calculated by the sum of the sample correlation function and a correction term: )( ) ∑n−k ( ˆ ˆ V − V V − V j tot j+k tot j=1 2k + . (7) ρˆk = ( ) 2 ∑n n−1 ˆ V − V j tot j=1 3
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We estimate the excitatory and inhibitory synaptic conductances after determining the total conductance and the effective reversal potential. The estimators follow from Eqs. (3) and (4): (8)
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gˆtot (Vˆtot − VI ) − gL (VL − VI ) − Iinj , VE − VI gˆI = gˆtot − gˆE − gL .
gˆE =
(9)
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In this paper, the parameters of the estimation model are Cm = 1.0 nF, gL = 0.1 µS, VL = −70 mV, VE = 0 mV, and VI = −75 mV. These parameters are assumed to be available, because it is possible to measure them by different experiments.
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2.2. Simulation model The leaky integrator model and a Hodgkin−Huxley type model were used to investigate the estimation performance of the conductance estimation method (see Section 2.1).
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2.2.1. Leaky integrator model The voltage V (t) of the leaky integrator model neuron is described by Eq. (1). The membrane parameters were set to Cm = 1.0 nF, gL = 0.1 µS, VL = −70 mV, VE = 0 mV, and VI = −75 mV. The injected current Iinj was zero, unless otherwise stated. The synaptic conductances were described by the point-conductance model (Destexhe et al., 2001), √ dgE 2 gE − g¯E =− + σE ξE (t), (10) dt τE τE √ dgI 2 gI − g¯I =− + σI ξI (t), (11) dt τI τI
where g¯E(I) and σE(I) are the mean and SD of the synaptic conductance, τE(I) = 0.5 (1.0) ms is the synaptic time constant, and ξE(I) (t) is the independent Gaussian white noise with zero mean and unit SD. The SD of the excitatory conductance was assumed to be proportional to the corresponding mean, σE = 0.2¯ gE (Miura et al., 2007). The mean and SD of the inhibitory conductance were g¯I = 50 nS and σI = 10 nS. Eqs. (1), (10), and (11) were solved numerically using the forward Euler integration method with a time step of 0.01 ms. 4
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dV dt
= −gL (V − VL ) − INa − IK
−gE (t)(V − VE ) − gI (t)(V − VI ) + Iinj .
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2.2.2. Hodgkin−Huxley type model We used a Hodgkin−Huxley type model of a cortical neuron proposed by Pospischil et al. (2008), which is biophysically realistic compared with the leaky integrator model (Eq. 1). The voltage V of the model neuron receiving excitatory and inhibitory inputs was described by
(12)
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The passive parameters (Cm , gL , VL , VE , and VI ) were the same as in the leaky integrator model (Section 2.2.1). The Na+ current INa was described by (13)
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INa = gNa m3 h(V − ENa ),
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dm = αm (V )(1 − m) − βm (V )m, dt −0.32(V + 45) αm (V ) = −(V +45)/4 , e −1 0.28(V + 18) βm (V ) = (V +18)/5 , e −1 dh = αh (V )(1 − h) − βh (V )h, dt αh (V ) = 0.128e−(V +41)/18 , 4 βh (V ) = , −(V 1 + e +18)/5
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where gNa = 50 µS, and ENa = 50 mV. The delayed-rectifier K+ current IK was described by IK = gK n4 (V − EK )
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dn = αn (V )(1 − n) − βn (V )n, dt −0.032(V + 43) , αn (V ) = −(V +43)/5 e −1 βn (V ) = 0.5e−(V +48)/40 , 5
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where gK = 5 µS, and EK = −90 mV. The synaptic conductances (gE (t), gI (t)) were described by the point conductance model (Eqs. 10 and 11), and the parameters were the same as in Section 2.2.1. This model was solved numerically using the forward Euler integration method with a time step of 0.01 ms. Fig. 1 shows the voltage traces generated by the two model neurons receiving excitatory and inhibitory inputs. Two levels of excitatory inputs are injected into these model neurons, and the voltage of both model neurons increases with the excitatory input. The Hodgkin−Huxley model occasionally generates spikes for the stronger input (Fig. 1b). In contrast, the leaky integrator model does not generate spikes, because it does not incorporate the spike generation mechanism.
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Figure 1: Simulated voltage traces for the weak (black) and strong (red) excitatory input. (a) Leaky integrator model. (b) A Hodgkin−Huxley type model. The model generates a spike at t = 0.15 s for the strong input. The voltage traces are shown for 0.4 s, which is the observation window used in our analysis.
2.3. Estimation errors Estimation errors of the excitatory and inhibitory conductances were evaluated by the mean relative error (MRE), M 1 ∑ gˆk − g¯ , MRE = M k=1 g¯
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and the mean absolute relative error (MARE),
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M 1 ∑ gˆk − g¯ MARE = , M k=1 g¯
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where gˆk is an estimate of the mean conductance in the k-th trial and g¯ is the true mean. MRE measures the estimation bias, whereas MARE measures the sum of the bias and the variability of an estimator. The errors were calculated from M = 1, 000 trials, unless otherwise stated.
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3. Results
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3.1. A single spike deteriorates the synaptic conductance estimation Simulated data were generated by injecting excitatory and inhibitory conductance inputs into the leaky integrator model (Section 2.2.1) and into a Hodgkin−Huxley type model (Section 2.2.2). Fig. 2 shows the effect of excitatory input on the conductance estimation errors evaluated by the mean relative error (MRE) and the mean absolute relative error (MARE). The errors for the leaky integrator model are almost constant over the whole range of excitation. The relative errors are negative, which indicates that the estimates are smaller than the true values. The two errors (MRE and MARE) are -10 % and 20 % for excitatory conductance, and -20% and 50 % for inhibitory conductance, respectively. For weak inputs (¯ gE ≤ 30 nS), the errors for the Hodgkin−Huxley model are also almost constant. As the excitatory input increases, the relative error decreases slightly and then increases sharply around g¯E ≈ 40 nS. The relative errors for the Hodgkin−Huxley model are positive for strong inputs, which indicates that the estimates are larger than the true values. The errors for strong inputs (¯ gE > 40 nS) are not shown in Fig. 2, because they are larger than 200 %. For weak inputs (¯ gE < 40 nS), the errors for the Hodgkin−Huxley model are comparable to those for the leaky integrator model. In contrast, the errors for the Hodgkin−Huxley model are much larger than those for the leaky integrator model for strong inputs (¯ gE > 40 nS). As shown in Fig. 1, the Hodgkin−Huxley model occasionally discharges spikes for the strong inputs. This result suggests that the spikes lead to a large increase in the errors of conductance estimation.
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Figure 2: Effect of excitatory input on the synaptic conductance estimation. (a, b) Dependence of the errors (a: MRE, b: MARE) for the excitatory conductance gE on the input conductance g¯E . (c, d) Dependence of the errors (c: MRE, d: MARE) for the inhibitory conductance gI on the input conductance g¯E . The errors for the leaky integrator model (LI: black) are compared with those of the Hodgkin−Huxley model (HH: red dashed).
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In the following, we focus on the estimation errors for the Hodgkin−Huxley model, because the effect of excitatory inputs on the estimation errors is modest for the leaky integrator model (Fig. 2). Fig. 3 shows the effect of spikes on the estimation errors. The errors increase immediately when the model starts to generate spikes. When the excitatory input is intermediate (40 < g¯E ≤ 45), the model generates spikes in only some trials. The estimation errors of no spike trials are much smaller than those of all the trials. Table 1 summarizes the errors for a sub-threshold input (¯ gE = 35 nS) and the errors for an intermediate input (¯ gE = 45 nS) where the average number of spikes was 1.0. Although the duration of a spike is very short (less than 1 % of the observation period), even a single spike markedly increases the errors. It is known that the Hodgkin−Huxley model exhibits nonlinear behavior even in the sub-threshold regime (Jolivet and Gerstner, 2004). The mean absolute errors of ”no spike” trials slightly increase with the excitatory input (Fig. 3 b, d), which is considered to be due to the sub-threshold nonlinearity. The result shows that the effect of this sub-threshold nonlinearity on the estimation accuracy is modest compared to the effect of a single spike (Fig. 3 and table 1). The reason may be that Equation (1) is rather good approximation of the Hodgkin-Huxley model in the sub-threshold regime but not during the spike itself. The effect of a single spike on the auto-correlation function of the voltage is shown in Fig. 4. The estimates of the total conductance from the voltage trace with a spike was 990 nS, whereas that with no spike was 120 nS. Thus, the synaptic conductance is largely over-estimated in the trials containing a spike. This result suggests that even a single spike significantly deteriorates the conductance estimation. Here, we considered only the increasing excitatory input. However, the result does not change quantitatively for the balanced input (Petersen et al., 2014; Kobayashi et al., 2015) where the excitatory and inhibitory input increase simultaneously (data not shown).
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Table 1: Estimation errors of synaptic conductance for the Hodgkin−Huxley model. Simulated data were generated from two levels of excitatory input (¯ gE = 35 and 45 nS). For the stronger input, the errors were calculated from no spike trials (¯ gE = 45 nS, no spikes) and from all the trials (¯ gE = 45 nS).
MRE (Inh) −39 −120 1, 100
MARE (Exc) MARE (Inh) 23 56 43 120 500 1, 200
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MRE (Exc) g¯E = 35 nS −16 g¯E = 45 nS, No spikes −43 g¯E = 45 nS 500
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Figure 3: Effect of spikes on the synaptic conductance estimation in the Hodgkin−Huxley model. (a, b) Dependence of the errors (a: MRE, b: MARE) for the excitatory conductance gE on the input conductance g¯E . (c, d) Dependence of the errors (c: MRE, d: MARE) for the inhibitory conductance gI on the input conductance g¯E . The mean errors of the trials with no spikes (gray) are compared with those of all the trials (red dashed). The arrows indicate the minimal input conductance required for generating spikes. (e, f) Dependence of the average number of spikes Nsp on the input conductance g¯E .
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Figure 4: Effect of a single spike on the auto-correlation function. (a) Simulated voltage trace with a spike (red) and in absence of spikes (blue). (b) Corresponding auto-correlation functions (ACF) using the same colors as in (a). The exponential fits to the ACF are shown in gray. The excitatory input was g¯E = 45 nS for both voltage traces.
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3.2. Approaches to estimate synaptic conductance in the firing regime We have shown that even a single spike causes significant deterioration in the synaptic conductance estimation. Here, we consider two approaches to reduce the deterioration effect. The first approach is to inject a negative current into the neuron to prevent spike discharges, which has been traditionally used in electrophysiological studies (Kobayashi et al., 2016). We confirm that the negative current injection markedly reduces the estimation errors of the synaptic conductances (Fig. 5, Table 2). The second approach is to remove spike waveforms from the voltage trace, which was suggested by Lansky et al. (2006) and Berg and Ditlevsen (2013). The synaptic conductances were estimated from sub-threshold parts of the voltage trace, namely V (t) < −40 mV. We again confirm that the spike removal also reduces the estimation errors (Table 2). The ameliorating effect of negative current injection is larger than the effect of spike removal.
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Table 2: Comparison of the approaches to recover from the deterioration in the conductance estimation caused by spikes. Estimation errors are compared among the three methods: estimation from the whole voltage trace (Original), estimation from the whole voltage trace with negative current injection Iinj = −1.0 nA (Current injection), and estimation from the sub-threshold voltage trace (Spike removal).
MRE (Inh) 1, 100 −26 −85
MARE (Exc) 520 15 31
MARE (Inh) 1, 200 57 85
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MRE (Exc) Original 500 Current injection −7 Spike removal −30
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Figure 5: Negative current injection can recover from the deterioration in the conductance estimation caused by spikes. (a, b) Dependence of the estimation errors (a: MRE, b: MARE) on the injected current Iinj in Eq. (12). Cyan and dashed magenta represent the errors of excitatory and inhibitory conductance, respectively. (c) Dependence of the average number of spikes on the injected current. The mean excitatory conductance was g¯E = 45 nS. The arrows indicate the minimal injected current required for generating spikes. Note that the negative current (Iinj < 0) was injected to prevent spike generation.
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4. Conclusions
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We have shown that a single spike deteriorates the estimation of synaptic conductance. The effect of a spike is much larger than the effect of subthreshold nonlinearity in the Hodgkin−Huxley model. We have also proposed two approaches to recover from this deterioration caused by spikes: negative current injection and spike removal. It is confirmed that these approaches dramatically reduced the estimation error. As a future work, we are planning to investigate how to remove spikes from a voltage trace for an accurate conductance estimation.
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Acknowledgements We thank two anonymous reviewers for their valuable comments. This work was supported by ACT-I, Japan Science and Technology Agency, JSPS KAKENHI Grant Number 25870915 and 15K06695, the Okawa Foundation for Information and Telecommunications and the open collaborative research and MOU grant at National Institute of Informatics in Japan. Supported by the Institute of Physiology RVO:67985823 and the Czech Science Foundation project 15-08066S.
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