Abstract

We consider a classical space-clamped Hodgkin-Huxley model neuron stimulated by synaptic excitation and inhibition with conductances represented by Ornstein-Uhlenbeck processes. Using numerical solutions of the stochastic model system obtained by an Euler method, it is found that with excitation only, there is a critical value of the steady-state excitatory conductance for repetitive spiking without noise, and for values of the conductance near the critical value, small noise has a powerfully inhibitory effect. For a given level of inhibition, there is also a critical value of the steady-state excitatory conductance for repetitive firing, and it is demonstrated that noise in either the excitatory or inhibitory processes or both can powerfully inhibit spiking. Furthermore, near the critical value, inverse stochastic resonance was observed when noise was present only in the inhibitory input process. The system of deterministic differential equations for the approximate first- and second-order moments of the model is derived. They are solved using Runge-Kutta methods, and the solutions are compared with the results obtained by simulation for various sets of parameters, including some with conductances obtained by experiment on pyramidal cells of rat prefrontal cortex. The mean and variance obtained from simulation are in good agreement when there is spiking induced by strong stimulation and relatively small noise or when the voltage is fluctuating at subthreshold levels. In the occasional spike mode sometimes exhibited by spinal motoneurons and cortical pyramidal cells, the assumptions underlying the moment equation approach are not satisfied. The simulation results show that noisy synaptic input of either an excitatory or inhibitory character or both may lead to the suppression of firing in neurons operating near a critical point and this has possible implications for cortical networks. Although suppression of firing is corroborated for the system of moment equations, there seem to be substantial differences between the dynamical properties of the original system of stochastic differential equations and the much larger system of moment equations.

1  Introduction

The stochastic nature of neuronal discharges was first reported in the 1930s, through experiments that found variability in the responses of frog myelinated axon to shocks of the same intensity and duration, notably by Monnier and Jasper (1932), Blair and Erlanger (1932), and Pecher (1936, 1937). Concerning Pecher, Verveen has documented an interesting account of his career, which started in Belgium and ended mysteriously at the age of 28 in the United States after researching radioactive substances whose nature was considered a military secret. (See www.verveen.eu, which also has links to Pecher’s articles, in French.) Some notable later contributions, including review, on this aspect of stochasticity in neurons are Verveen (1960), Lecar and Nossal (1971), and Clay (2005).

Since these early discoveries, a large number of experimentalist studies has revealed the stochastic nature of nearly all neuronal activity, the latter term referring mainly to action potential generation. This embraces single neurons and their component parts and populations of neurons and glia. Some of the pioneering works were on interspike interval variability in muscle spindles (Brink, Bronk, & Larrabee, 1946), cells in the auditory system (Gerstein & Kiang, 1960), and those in the visual cortex (Burns & Webb, 1976).

Mathematical modeling of the complex dynamical processes underlying such activity has since flourished (Bachar, Batzel, & Ditlevsen, 2013). Much of the theoretical work has focused on channel noise at both the single channel level (Colquhoun & Hawkes, 1981) and the patch level, modeled by a diffusion limit for a birth and death process (Tuckwell, 1987). The role of such noise in the generation of action potentials—for example by altering the firing threshold—has been explored in (for example) White, Rubinstein, and Kay (2000), Austin (2008), and Li, Schmid, Hänggi, and Schimansky-Geier (2010).

Models with linear subthreshold behavior and synaptic noise have been studied since the 1960s. There have been more recent analyses of them, such as those of Hillenbrand (2002), Ditlevsen and Lansky (2005, 2007), and Berg and Ditlevsen (2013). Most nonlinear electrophysiological models of neuronal activity are based on Hodgkin-Huxley (HH) type systems (Hodgkin-Huxley, 1952) and include spatial models (Horikawa, 1991; Sauer & Stannat, 2016). Some recent studies include those of Wenning, Hoch, and Obermayer (2005), who employed a model similar to that explored in this letter, and Finke, Vollmer, Postnova, and Braun (2008), who included both additive noise and channel noise in the form of Ornstein-Uhlenbeck processes (OUPs) for some activation variables, and Yi et al. (2015), who studied noise effects on spike threshold in a two-dimensional HH-type model with synaptic noise represented by an OUP.

The focus of this letter is on the classical HH system with synaptic noise and its analysis by the moment method, in a similar vein to that for the same system with additive white noise (Rodriguez & Tuckwell, 2000; Tuckwell & Jost, 2009). With weak additive white noise or with conductance-based noise as in the present model (Tuckwell, Jost, & Gutkin, 2009) or with colored noise (Guo, 2011), and mean input currents near the threshold for repetitive firing, the firing rate undergoes a minimum (inverse stochastic resonance) as the noise level increases from zero. We examine the responses of the system to synaptic input, either excitatory alone or with inhibition near the threshold for repetitive spiking, and compare solutions obtained by the moment method with simulation. The moment method and extensions of it have been employed in several recent studies of neuronal networks (Deco & Marti, 2007; Marreiros, Daunizeau, Kiebel, & Friston, 2008; Hasegawa, 2009, 2015; Deco et al., 2013; Franović, Todorović, Vasović, & Burić, 2013) and genetic networks (Sokolowski and Tkačik, 2015).

2  The HH Equations with Random Synaptic Currents

The standard space-clamped HH system consists of four ordinary differential equations for the electrical potential , the potassium activation variable , the sodium activation variable , and the sodium inactivation variable . The latter three variables take values in , and their differential equations involve the coefficients , which depend on V. In a previous article (Tuckwell & Jost, 2009), the HH system with an additive current, including deterministic and random components, was analyzed. Here we use similar techniques for the HH system with random synaptic excitation and inhibition. (See also Tuckwell & Jost, 2012, and Bashkirtseva, Neiman, & Ryashko, 2015, for further analysis of the effects of noise in the HH model with additive noise.)

Random synaptic inputs can be included in several ways (e.g., Rodriguez & Tuckwell, 1998), but we restrict our attention to cases where there are no discontinuities in V but rather the synaptic currents can be represented with diffusion processes, which makes the mathematical formalism somewhat less complicated. The resultant system consists of six coupled stochastic differential equations (SDEs) where the stochasticity is explicit only in the fifth and sixth equations.

The form of the system under consideration here is that of an n-dimensional temporally homogeneous diffusion process where the jth component satisfies the stochastic differential equation,
formula
2.1
where the Wks are independent standard (zero mean, variance t at time t) Wiener processes and it is assumed that conditions for existence and uniqueness of solutions are met (Gihman & Skorohod, 1972).

2.1  Description of the Model

For our model, we let the first four variables be , , , and , and be the vector of all six components. The first equation of a general HH system with synaptic inputs can be written as
formula
2.2
where Isyn represents the synaptic input currents and we have defined the sum of the standard HH currents as
formula
2.3
where , , and gL are the maximal (constant) potassium, sodium, and leak conductances per unit area with corresponding equilibrium potentials VK, VNa, and VL, respectively. The second, third, and fourth equations take the standard HH forms:
formula
2.4
formula
2.5
formula
2.6
In the following, we denote the derivatives of by
A representation of certain synaptic inputs has been successfully used in models for the spontaneous random spiking of cat and rat neocortical pyramidal neurons (Destexhe, Rudolph, Fellous, & Sejnowski, 2001; Fellous, Rudolph, Destexhe, & Sejnowski, 2003) in which the excitatory and inhibitory inputs constituting the ongoing background input are presumed to be composed mainly of small and frequent glutamatergic and GABA-ergic postsynaptic currents, which could be separately identified. A similar model was employed for an HH neuron with conductance-driven input by Tuckwell et al. (2009) in a demonstration of the robustness of the phenomenon of inverse stochastic resonance. In the Destexhe et al. (2001) model, there is, in addition to the usual three HH currents, an M-type potassium current, which, as in Tuckwell et al. (2009), is not included here. In actual neocortical pyramidal cells, there are several more component currents than in the Destexhe et al. (2001) model (see, e.g., Yu, Shu, & McCormick, 2008, for a partial list). The overall synaptic current is then written by Destexhe et al. (2001) as
formula
2.7
where are equilibrium potentials for excitatory and inhibitory synaptic input, the synaptic excitatory and inhibitory conductances at time t are and , respectively, and A is the “total membrane area.” The synaptic conductances are ascribed the stochastic differential equations
formula
2.8
formula
2.9
where are time constants, are equilibrium values, are corresponding (assumed independent) standard Wiener processes, and are noise amplitudes. Thus, the processes ge and gi are of the OUP type.
In this model, the first of the six SDEs is simply
formula
2.10
with
formula
2.11
where and .
For X5 and X6, we have
formula
2.12
and
formula
2.13
where we identify and and where
formula
2.14
formula
2.15
Further, and , and these are the only two nonzero gjks in this model so in equation 2.1. There are 27 distinct first- and second-order moments for this model; differential equations for their approximations are obtained in section 4.

3  Simulation Results

Simulated solutions of the system of six coupled SDEs equations defining the above HH model neuron with random synaptic inputs are obtained with a simple Euler method in which discretization is applied with a time step of . Unless stated otherwise, the value is employed in the calculations described below.

3.1  Excitation: Critical Value and Choice of

With no inhibition and no noise so that , there is a critical value of for repetitive firing. When repetitive (presumed continuing indefinitely) firing does not occur for any value of tested. Figure 1A shows the voltage response without noise and with . There is only one spike at about . With very small noise , the results are shown in Figure 1B for 10 trials, the response being almost identical in each trial. When the noise amplitude is greater at as in Figure 1C, second spikes may emerge as is the case in 4 of the 10 trials depicted. Figure 1D shows 4 voltage trajectories for a still greater noise level . Here there are in all cases (many not shown) an apparently unceasing sequence of randomly occurring spikes, despite the fact that the value of is less than the critical value for repetitive firing without noise.

Figure 1:

Simulation results with excitation only and various noise levels. In all cases, steady-state excitation below the critical value for repetitive firing and a time step of . (A) No noise (1 trial) giving one spike. (B) Ten trials with noise . One spike only, being practically the same on each trial. (C) Ten trials with . The noise is sufficient to give rise to a second spike in 4 of the 10 trials. (D) Four individual trials with . In all cases examined, multiple sustained spiking at random times occurs.

Figure 1:

Simulation results with excitation only and various noise levels. In all cases, steady-state excitation below the critical value for repetitive firing and a time step of . (A) No noise (1 trial) giving one spike. (B) Ten trials with noise . One spike only, being practically the same on each trial. (C) Ten trials with . The noise is sufficient to give rise to a second spike in 4 of the 10 trials. (D) Four individual trials with . In all cases examined, multiple sustained spiking at random times occurs.

Using the previously found (Tuckwell et al., 2009) critical value as a guide, spike trains were examined for values of close to that value for various values of . Results will be reported only for the two values and . For , there were 4 and 6 spikes, respectively, for the smaller and larger time steps, and for the slightly larger value , there were 4 spikes of declining amplitude for the smaller time step, whereas for the larger time step, there were 13 spikes in 240 ms whose amplitudes finally remained constant (see Figure 2A). With , there were 7 spikes of declining amplitude for the smaller time step and an apparently repetitive train for the larger time step.

Figure 2:

Spike trains for just above the critical value for repetitive spiking, showing the effects of smaller (red) and larger (black) time steps of 0.002 and 0.015, respectively. The larger time step leads to a higher frequency. The horizontal segment indicates stable spike amplitudes for both time steps.

Figure 2:

Spike trains for just above the critical value for repetitive spiking, showing the effects of smaller (red) and larger (black) time steps of 0.002 and 0.015, respectively. The larger time step leads to a higher frequency. The horizontal segment indicates stable spike amplitudes for both time steps.

Finally, with just above the previously determined critical value, an apparently stable repetitive spike train was obtained with both the larger and smaller time step. This is seen in Figure 2 and suggests that the critical value is very close to 0.1125. In general, it was observed that the larger time step sped up the spiking and tended to make it more stable. However, the results indicated that it is preferable to use the smaller time step although it leads to significantly greater computation times.

3.2  Inhibition by Noise with Excitation Only

With excitation only at , so that without noise there is repetitive periodic firing, noise of a small amplitude can lead to a greatly reduced number of spikes. This is illustrated in Figure 3, where the three columns show four trials, of length 100 ms, for each of three values of , being 0.0025, 0.01, and 0.025, increasing from left to right. With no noise, there are six spikes (see Figure 2). The smallest of the values of leads to the greatest reduction in average spike numbers, to 3.0, whereas the largest value of has an average spike number of 5.75.

Figure 3:

Spike trains for just above the critical value for repetitive spiking, showing the effects of small noise of three different magnitudes, increasing from left to right, in four trials.

Figure 3:

Spike trains for just above the critical value for repetitive spiking, showing the effects of small noise of three different magnitudes, increasing from left to right, in four trials.

3.3  Inhibition by Noise with Excitation and Inhibition

In previous investigations of the inhibitory effects of noise on repetitive firing induced by synaptic input (Tuckwell et al., 2009), only excitatory inputs have been considered. Here we briefly consider a few cases in which the synaptic input is both excitatory and inhibitory and there is noise in either the excitatory or inhibitory component or both. First, it was required to find a combination of excitation and inhibition that would lead to repetitive spiking in the deterministic case. Without a formal proof, it seems that for any level of inhibition , there can always be found a level of excitation to give repetitive spiking. With , close to the critical value of for excitation only, spike trains were examined with a time step of 0.015 for various values of . The elicited trains for and did not exhibit repetitive spiking, as shown in the first two panels in the top row of Figure 4. When was increased slightly to 0.1790, as in the right-hand panel of the first row, repetitive spiking was sustained, indicating that this level of excitation was critical for the given level of inhibition. In the remaining three rows of Figure 4 are shown the resulting spike trains with noise for three trials in which the values of and are those that gave repetitive spiking without noise. In the second row, there is noise of a small magnitude in the excitation only, whereas in the third row, noise of magnitude is present in the inhibitory input only. Finally, in the fourth row, there is noise in both the excitatory and inhibitory input processes with , such that the sum of the amplitudes is the same as in rows 2 and 3. The percentage reductions in average spike numbers, for these (small) sample sizes, are 78, 67, and 64 for noise in excitation, inhibition, and both, respectively. This preliminary investigation indicates that inhibition of repetitive spiking by noise is as strong when there is inhibition present as when there is excitation only and occurs regardless of whether the noise arises in excitatory, inhibitory, or both input processes.

Figure 4:

(A) In the top three records, there is no noise but both excitatory and inhibitory inputs. In the leftmost panel, with and , and in the middle panel, with and , repetitive firing is not established. When, as in the right-hand panel, the larger value is employed with the same value of , repetitive firing occurs. (B) In the results of the second row, the values of and are as in the right-hand panel of the first row, leading to repetitive spiking in the absence of noise, but now a small noise is added only to the excitatory component, giving rise to a large degree of inhibition. (C) As in panel B, but now noise with amplitude is added to only the inhibitory component, which also inhibits the spiking. (D) As in panels B and C, except that the noise is spread equallly among excitatory and inhibitory inputs with and . Significant inhibition of spiking is observed in this case also.

Figure 4:

(A) In the top three records, there is no noise but both excitatory and inhibitory inputs. In the leftmost panel, with and , and in the middle panel, with and , repetitive firing is not established. When, as in the right-hand panel, the larger value is employed with the same value of , repetitive firing occurs. (B) In the results of the second row, the values of and are as in the right-hand panel of the first row, leading to repetitive spiking in the absence of noise, but now a small noise is added only to the excitatory component, giving rise to a large degree of inhibition. (C) As in panel B, but now noise with amplitude is added to only the inhibitory component, which also inhibits the spiking. (D) As in panels B and C, except that the noise is spread equallly among excitatory and inhibitory inputs with and . Significant inhibition of spiking is observed in this case also.

Further results were obtained with the levels of excitation and inhibition and which led to repetitive spiking as in Figure 4A. Fifty trials of length 100 ms were performed for 30 values of the inhibitory noise parameter , from 0 to 0.1, with no noise in the excitatory process. The mean number of spikes versus noise level is shown in Figure 5 (black circles), and it is seen that the spike rate undergoes a minimum around values of just less than 0.02. Away from the critical value for repetitive spiking, when the level of the excitatory input was increased to but with the same value of , the firing rate undergoes a much weaker minimum as is increased from 0 to 0.1. These results parallel those obtained previously for increasing the excitatory noise level (Tuckwell et al., 2009). Somewhat surprisingly, therefore, the phenomenon of inverse stochastic resonance occurs with increasing level of inhibitory noise alone.

Figure 5:

The mean number of spikes per trial is plotted against the inhibitory noise parameter , with no noise in the excitatory process, for values of and near a critical value for repetitive spiking (black circles) and at a level of excitation above the critical value (red diamonds). For the first set, a pronounced minimum occurs in the firing rate near , providing evidence of inverse stochastic resonance with respect to inhibitory noise alone.

Figure 5:

The mean number of spikes per trial is plotted against the inhibitory noise parameter , with no noise in the excitatory process, for values of and near a critical value for repetitive spiking (black circles) and at a level of excitation above the critical value (red diamonds). For the first set, a pronounced minimum occurs in the firing rate near , providing evidence of inverse stochastic resonance with respect to inhibitory noise alone.

4  Differential Equations for the Approximate First- and Second-Order Moments

We will find deterministic differential equations satisfied by the approximations for the means, variances, and covariances of the components of the n-component vector-valued random process in the above model, using the scheme of Rodriguez and Tuckwell (1996). Accordingly, for relatively small noise amplitudes, the exact means and covariances may sometimes be approximated by the functions , and , respectively, which obey a system of deterministic differential equations.

The vector of means at time t is denoted by . These quantities are found to satisfy the following systems of ordinary differential equations,
formula
4.1
whereas the covariances are determined by
formula
4.2

In our model, this equation is simpler because the last line of equation 4.2 is absent, there being no triple-sum contribution involving second derivatives or products of first-order derivatives of the gjks because such terms are all constants, most of which are zero.

In order to simplify the notation in the following equations we give some definitions. Let
formula
and let
formula
4.3

4.1  The Means

Evaluating the required first- and second-order partial derivatives of , we obtain the following differential equations for the means , of , respectively. For the voltage, we have
formula
For the auxiliary variables, we have
formula
For the means of the synaptic conductances, we have
formula
Since these are means of OUPs, exact solutions are known, which in fact coincide with the approximations. Solutions are
formula
and likewise for .

4.2  The Variances

For the above model of an HH system with random synaptic input, the differential equations for the variances of , obtained by substituting the appropriate derivatives and coefficients, are as follows. For the variance of V, we have:
formula
For the variances of n, m, and h, we find
formula
For the variances of the excitatory and inhibitory conductances, we find
formula
Since these variances are for OUPs, we know the exact solutions,
formula
and likewise for , coinciding with the approximations.

4.3  Remaining Covariances

The remaining 15 covariances are solutions of the following differential equations:
formula
However, the correct solution of the last equation must be since the Wiener processes We and Wi are independent. Hence, the terms in H2 in the equations for C15 and C16 become and .

5  Comparison of Results for Moment Equations and Simulation

Generally agreement between the results for the moment equation method (MEM) and simulation, in the sense that the mean and variance of the voltage as functions of time were reasonably close, was obtained when the net mean driving force was large and the variances small. This is to be expected from the assumptions under which the moment equations are derived.

Figure 6 shows one such case where there is excitation only with parameter values for synaptic input, , , , , with 50 trials and standard HH parameters as given in the appendix.

Figure 6:

An example, with excitation only, of the calculated mean and variance of the voltage as functions of time of the voltage where there is good agreement between results for simulation (red) and moment equations (blue). The input parameters are and , with no inhibition and standard HH parameters. The means for the two methods are indistinguishable.

Figure 6:

An example, with excitation only, of the calculated mean and variance of the voltage as functions of time of the voltage where there is good agreement between results for simulation (red) and moment equations (blue). The input parameters are and , with no inhibition and standard HH parameters. The means for the two methods are indistinguishable.

For most input parameter sets examined, the maximum variance for the MEM was greater than that obtained by simulation. In a very few examples, particularly for large and with a very small time step, the maximum variance for the simulation solution was greater than that for the MEM. The approximation for the variance could be overestimating the true variance, but it could also reflect that either there is a very small probability that the first spike would be delayed, advanced, or even entirely inhibited, leading to an increased variance, or that the amplitude in rare cases is larger during the first spike, also increasing the variance, but these probabilities are so small that they are not seen in practice in the simulations.

In a second example, with reasonable agreement for the two methods, there is synaptic excitation and synaptic inhibition with parameters , , , , , , , and standard HH parameters as given in the appendix. Thirty simulation runs of 50 trials each were performed. Again, the mean of obtained by each method was almost identical, as seen in the top part of Figure 7, where there is one blue curve for the MEM and 30 red curves for the simulations.

Figure 7:

An example, with both excitation and inhibition, of the calculated mean and variance of the voltage as functions of time of the voltage where there is fairly good agreement between results for simulation (red) and for moment equations (blue). The timescales are different for the mean and the variance. The principal synaptic input parameters are , , , . For remaining parameters, see the text. Results are given for 30 runs of 50 trials each. Note the variability in the curves for the variance by simulation (lower figure, red curves). The means for the two methods are practically indistinguishable.

Figure 7:

An example, with both excitation and inhibition, of the calculated mean and variance of the voltage as functions of time of the voltage where there is fairly good agreement between results for simulation (red) and for moment equations (blue). The timescales are different for the mean and the variance. The principal synaptic input parameters are , , , . For remaining parameters, see the text. Results are given for 30 runs of 50 trials each. Note the variability in the curves for the variance by simulation (lower figure, red curves). The means for the two methods are practically indistinguishable.

The corresponding results for the variance of are shown in the bottom part of Figure 7. For the MEM, the maximum variance of is 0.0015, which is greater than the maximum variance in each of the 30 simulation runs. For the latter, the maximum value of all the maxima is 0.0014, which is fairly close to the value for the MEM, and the minimum value is 0.00069, with an average of 0.0010, which is about 33% less than the MEM value. For 29 of the 30 runs, simulation values of Var were less than the MEM value for all t, and in the remaining case, the simulation value crossed the MEM value just after the peak value and remained above it for most of the falling phase.

5.1  Occasional Spike Mode

The spiking reported in Destexhe et al. (2001) and Fellous et al. (2003) is classified as being from a cell operating in the occasional spike mode (Calvin, 1975), a term first used with reference to spinal motoneuron spiking. There is insufficient net depolarizing current to give rise to a sustained train of action potentials, but occasional large excursions to suprathreshold states arise due to random synaptic input.

With a small depolarizing current of , which is less than the critical value for repetitive firing, and with the standard set of synaptic input parameters reported in column 1 of Table 1 in Destexhe et al. (2001), S, S, , , but with , , and with initially resting conditions, a single sample path was generated, as depicted in the top left panel of Figure 8. A spike emerged that attained a maximum depolarization of about 97 mV at approximately 8.5 ms with corresponding sample paths for ge and gi shown up to 20 ms in the top right panel of Figure 8. Subsequently there are subthreshold fluctuations, labeled Vs for , about a mean of 2.84 mV depicted in the lower left panel of the figure. A histogram of voltage values for Vs from to and corresponding histograms of values of the excitatory and inhbitory conductances are also shown, similar to those in Figure 2 of Destexhe et al. (2001).

Figure 8:

(Top left) A spike followed by fluctuations in a steady state up to 100 ms. (Top right) The sample paths of ge and gi for the first 20 ms. After the voltage fluctuations are designated Vs with sample path and histogram of values shown in the first two bottom figures. (Bottom right) The histograms of ge and gi during the period from 40 to 100 ms. The latter are comparable to those in Figures 2 and 3 of Destexhe et al. (2001). For parameter values, see text.

Figure 8:

(Top left) A spike followed by fluctuations in a steady state up to 100 ms. (Top right) The sample paths of ge and gi for the first 20 ms. After the voltage fluctuations are designated Vs with sample path and histogram of values shown in the first two bottom figures. (Bottom right) The histograms of ge and gi during the period from 40 to 100 ms. The latter are comparable to those in Figures 2 and 3 of Destexhe et al. (2001). For parameter values, see text.

Starting with the values of all components V, n, m, h, ge, and gi at (beginning of Vs), three sets of simulations and moment equation calculations were performed with the results to 5 ms shown in Figure 9. There is good agreement between the means and variances determined by the two methods.

Figure 9:

The mean and variance for the subthreshold voltage fluctuations of the previous figure calculated by the MEM (blue curves) and three sets of simulations (red, black, and green curves).

Figure 9:

The mean and variance for the subthreshold voltage fluctuations of the previous figure calculated by the MEM (blue curves) and three sets of simulations (red, black, and green curves).

Figure 10 illustrates further the occurrence of the occasional spike mode over a time period of 400 ms. In all three records, the inhibitory steady-state conductance is the standard (Destexhe et al., 2001, Table 1, column 1) value of , and its noise amplitude is the standard value of . In Figure 10A, there are no spikes with , at four times the standard value of 0.012 and with , which is the standard value. In the middle record of Figure 10B, there is a slightly less additive depolarizing drive with , less mean synaptic excitation with at 3.6 times the standard value but with considerably greater noise amplitude of 3.2 times the standard value, which is sufficient to induce occasional spiking. In Figure 10C, there is an initial singlet spike followed by two doublets with intradoublet intervals of about 20 ms. Here the parameters are all as in Figure 10A (no spikes), but the excitatory noise level is 3.25 times the standard value. With occasional spikes, it is clear that the moment method will not be suitable because the condition of a symmetric distribution of component values is not met.

Figure 10:

Sample paths of duration 400 ms with added current of strength and synaptic excitation and inhibition. (A) Only subthreshold fluctuations occur for this period, with , , and . (B) Occasional spike mode in which singlet spikes occur. Here , , and . (C) Occasional spike mode in which spikes occur, sometimes in pairs. Here , , and . Remaining parameters are held fixed. See the text.

Figure 10:

Sample paths of duration 400 ms with added current of strength and synaptic excitation and inhibition. (A) Only subthreshold fluctuations occur for this period, with , , and . (B) Occasional spike mode in which singlet spikes occur. Here , , and . (C) Occasional spike mode in which spikes occur, sometimes in pairs. Here , , and . Remaining parameters are held fixed. See the text.

5.2  Input Parameters of Destexhe et al. (2001)

As pointed out in the model description, Destexhe et al. (2001) studied a point HH model augmented with an M-type potassium current and random excitatory and inhibitory synaptic inputs. Apart from the M-type potassium current, their model has the same structure as the one employed here.

It was of interest to see how results for the MEM compared with those for simulation, despite the lack of knowledge of the complete set of parameters for the transient sodium and delayed rectifier potassium current, as well as the omission of the M-type current. We use data in column 1 of Table 1 of Destexhe et al. (2001) on properties of a layer 6 pyramidal cell (P-cell) of cat neocortex. A resting potential given in the text was stated to be −80 mV.

The P-cell model parameters are as follows. In equation 2.11 we need C, the whole cell capacitance, which, based on the given membrane area of 34,636 m, is 0.34636 nF. If conductances are in S, and voltages are in mV, then currents are in nA. The equilibrium excitatory and inhibitory conductances for the whole cell are given as S, S with reversal potentials of mV and mV relative to a resting value of mV. The leak conductance, using the value in Destexhe et al. (2001), is S, which is about six times less than the value obtained if the standard HH value of 0.3 mS/cm is employed. For the values of and , we use the data of Paré, Lang, and Destexhe (1998) to obtain 3.4636 S and 2.4245 S, respectively.

Results of a simulation for the P-cell model, including synaptic input and with HH activation and inactivation dynamics and no added current, are shown in Figure 11. The records have a duration of 100 ms during which the voltage, shown in the top left panel, fluctuates about a mean of 0.17 mV (above rest) with a standard deviation of 1.6 mV. The distribution of V is indicated by a histogram in the top right panel of Figure 11. The lower two panels show the time courses of the excitatory and inhibitory conductances, ge and gi. There are no spikes as the fluctuations do not take V to threshold values. The mean in Destexhe et al. (2001) is much higher so that spikes do arise occasionally—referred to as the occasional spike mode. The values of ge in Figure 11 are between about 0.001 and 0.023 S; those of gi are between −0.02 S and a maximum of about 0.09 S. That the conductance may become slightly negative is a minor deficiency of the model equations, being due to the fact that the conductances are unrestricted OUPs. (This was allowed for in Tuckwell et al., 2009.) Overall, the conductance fluctuations are comparable to those in Destexhe et al. (2001), being governed by the same stochastic differential equations.

Figure 11:

Properties of one sample path for the P-cell with excitatory and inhibitory synaptic input. (Top left) V in mV versus t in ms. (Top right) A histogram of the values of V over the 100 ms period. (Lower two panels) The corresponding sample paths for the excitatory and inhibitory conductances. For parameters, see the text.

Figure 11:

Properties of one sample path for the P-cell with excitatory and inhibitory synaptic input. (Top left) V in mV versus t in ms. (Top right) A histogram of the values of V over the 100 ms period. (Lower two panels) The corresponding sample paths for the excitatory and inhibitory conductances. For parameters, see the text.

The P-cell model with HH activations and inactivation can be made to fire by either introducing an additive depolarizing current or increasing the ranges of the fluctuations of the synaptic conductances.

Figure 12 shows two such sets of results where results for simulation compare favorably with those obtained by solving the moment differential equations. In both cases, there is an added depolarizing current  A/cm. For Figure 12A, the synaptic steady-state conductances are , with corresponding standard deviation parameters and , as above. After a broad small-amplitude spike, a steady state is attained at a depolarized level with no further spikes. The agreement of the MEM and simulation for the mean and variance of V is very good throughout the spike and immediately afterward. The variance rises and falls during the rising and falling phases of the spike. In Figure 12B, the steady-state synaptic conductances have the smaller values , , with larger corresponding standard deviation parameters and .

Figure 12:

(A) An example for the P-cell model with an added depolarizing current and strong synaptic excitation and inhibition such that threshold for spiking is exceeded. Here, there is very good agreement between the mean and variance of determined by simulation (red curves) and the MEM (blue curves). (B) A second example for the P-cell model with the same depolarizing current as in panel A but with weaker but noisier synaptic excitation and inhibition. There are four sets of simulations with the same number of trials. There is broad agreement between the variance versus time curves from simulation (various colors) and the result from MEM (blue curve). Note the secondary maximum in the variance. For both panels A and B, the means for the two methods are practically indistinguishable. For parameter values, see the text.

Figure 12:

(A) An example for the P-cell model with an added depolarizing current and strong synaptic excitation and inhibition such that threshold for spiking is exceeded. Here, there is very good agreement between the mean and variance of determined by simulation (red curves) and the MEM (blue curves). (B) A second example for the P-cell model with the same depolarizing current as in panel A but with weaker but noisier synaptic excitation and inhibition. There are four sets of simulations with the same number of trials. There is broad agreement between the variance versus time curves from simulation (various colors) and the result from MEM (blue curve). Note the secondary maximum in the variance. For both panels A and B, the means for the two methods are practically indistinguishable. For parameter values, see the text.

Again, a small-amplitude broad spike forms, after which a fluctuating steady state is attained. Interesting is the fact that in this case, the variance has two maxima: it rises on the leading edge of the spike, decreases around the peak, rises again until about halfway down the falling edge, and then declines to a steady-state value. This behavior of the variance is confirmed by the calculation by MEM.

6  Discussion and Conclusion

The Hodgkin-Huxley (1952) mathematical model for voltage and current responses in squid axon has formed a cornerstone for describing the dynamics of neuronal subthreshold and spiking activity in a large number of types of neurons in many parts of the nervous system in diverse species. Whereas the original model involved only three types of ionic current—leak, potassium, and sodium—models of most neurons have been found to require several types of sodium, potassium, and calcium currents, each of which has a distinct role in determining a cell’s electrophysiological properties. The HH system itself is an important starting point in the analysis of neuronal behavior and is used in this letter to examine the effects of random synaptic activity on repetitive spiking. Many of the findings are expected to carry over to more complex models.

This letter contains an extension of some previous studies of the effects of noise in space-clamped HH models. First are the results of the inclusion of additive white noise and excitatory synaptic noise represented by OUPs, on periodic spiking (Tuckwell et al., 2009). Second is the use of a system of deterministic differential equations to find approximately the first- and second-order moments of the HH variables (Rodriguez & Tuckwell, 2000; Tuckwell & Jost, 2009). In the first study, it was found that at input signals just greater than critical values for repetitive (limit cycle) spiking, weak noise could cause a substantial decrease in firing rate and that a minimum in firing rate occurred as the noise level increased from zero. In the second type of study, the system of ordinary differential equations for the approximate moments was solved and the results compared with simulations.

The model employed here includes synaptic input of the type supported by experimental analysis and modeling of the voltage response of cat and rat neocortical neurons (Destexhe et al., 2001; Fellous et al., 2003). The spiking observed in those experiments was occasional (Calvin, 1975), and the inputs were described as fast with small amplitude so that they could be represented by continuous random processes (OUPs). We have investigated the effects of these types of excitatory and inhibitory random inputs on repetitive spiking in the HH model. However, it is likely that similar qualitative results would obtain if the random synaptic inputs were purely additive.

The effect of weak random excitatory synaptic input, investigated in section 4.1.2, was to inhibit spiking induced by a purely excitatory current near the threshold for repetitive firing, confirming results in section IV of Tuckwell et al. (2009). Section 4.1.3 contained results for the effects of noisy synaptic input on repetitive spiking induced by the simultaneous application of steady excitatory and steady inhibitory currents. Here it was demonstrated (see Figure 4) that weak noise in either the excitatory or inhibitory input process or in both could strongly inhibit spiking. Thus, a small amount of noise in any synaptic input channel can cause a dimunition of repetitive firing near threshold. Of considerable interest were the findings that with both steady excitatory and inhibitory current, with no noise in the excitatory input, increasing the noise only in the inhibitory component from zero gave rise to a minimum in the firing rate as in the phenomenon of inverse stochastic resonance previously demonstrated for the HH model with additive noise and synaptic excitation only (see Figure 5).

With regard to inverse stochastic resonance, it is interesting to ask what a minimal type of model is that will possess this property. The phenomenon of inverse stochastic resonance in the HH model happens because the stable limit cycle (corresponding to repetitive firing) coexists with a stable fixed point (corresponding to a silent neuron), separated by an unstable limit cycle. This occurs in many conductance-based neuron models, such as the HH model, but also simpler models, such as the two-dimensional Morris-Lecar model for certain parameter values (see Ditlevsen & Greenwood, 2013). At least two dimensions are necessary to obtain limit cycles. If the stable and the unstable limit cycles are close and the process is on the stable limit cycle, small noise amplitudes might make the process cross the unstable limit cycle. If the process is then dragged to the stable fixed point by the flow, and this fixed point is far from the unstable limit cycle, the small noise amplitude might not be enough to push the system back to the stable limit cycle. Thus, small noise suppresses the oscillations. A minimal system to present such behavior is the following:
formula
6.1
formula
6.2
For and , and expressed in polar coordinates, and , the behavior is easy to understand:
formula
6.3
formula
6.4

There is a stable fixed point at , corresponding to , an unstable limit cycle for , and a stable limit cycle for . Letting biases the unstable limit cycle to resemble neuron models, where the unstable limit cycle is often close to the stable limit cycle only at some point in state space, where the crossing typically occurs. The model is illustrated in Figure 13 for , where the stable limit cycle is indicated with a solid magenta curve, the unstable limit cycle is the dashed magenta line, and the fixed point is the magenta point. Four example traces are simulated, of which two are being suppressed. To the right, there is a zoom of the region indicated in blue on the left panel, where the behavior along the trajectories close to the crossing point between attractors can be appreciated. For this level of noise, the crossing will, with high probability, happen where the two cycles are close; once the trajectory gets close to the fixed point, it cannot escape.

Figure 13:

To the right the process is simulated four times for . A full oscillation occurs twice, corresponding to a spike, and twice the oscillation is suppressed. The solid magenta line is a limit cycle, which is stable when there is no noise, the dashed magenta line is an unstable limit cycle, and the magenta dot is a stable equilibrium. To the right is a zoom of the blue square to the left, where the details of the crossings of the unstable limit cycle can be appreciated for the four simulated traces.

Figure 13:

To the right the process is simulated four times for . A full oscillation occurs twice, corresponding to a spike, and twice the oscillation is suppressed. The solid magenta line is a limit cycle, which is stable when there is no noise, the dashed magenta line is an unstable limit cycle, and the magenta dot is a stable equilibrium. To the right is a zoom of the blue square to the left, where the details of the crossings of the unstable limit cycle can be appreciated for the four simulated traces.

To test for inverse stochastic resonance, we simulated 1000 trajectories of four time units with integration time steps of 0.001, starting at , which lies on the stable limit cycle, for each value of from 0.05 to 1.2 in steps of 0.05, and recorded if the oscillation had been suppressed or not. In Figure 14, the proportion of trials out of the 1000 trials where the oscillation persisted is plotted against the value of . There is a clear minimum around , showing the phenomenon of inverse stochastic resonance.

Figure 14:

The proportion of trials out of 1000 where there was a first oscillation is plotted against the noise parameter . A pronounced minimum occurs near , providing evidence of inverse stochastic resonance in this schematic example.

Figure 14:

The proportion of trials out of 1000 where there was a first oscillation is plotted against the noise parameter . A pronounced minimum occurs near , providing evidence of inverse stochastic resonance in this schematic example.

Another pertinent issue is the possible role of noise, in both excitatory inputs and inhibitory inputs, in having a controlling influence on network activity, such as occurs in certain parts of the neocortex. As we saw above from Destexhe et al. (2001), neurons in such networks often operate spontaneously in the occasional spike mode, which points to their being below and possibly near a critical point for repetitive firing. If the noisy input is relatively weak, then it could act in an inhibitory fashion on single neurons and hence, in turn, on the networks containing them, leading to a considerable silencing of their activity.

In an experimental study of neuronal conductance fluctuations in sleep and wake periods in the cat parietal cortex, Rudolph, Pospischil, Timofeev, and Destexhe (2007) found that in most cells, inhibition was dominant and that spiking tended to be preceded by a drop in conductance, which indicated that the trigger for spiking was not an increase in excitatory input but a decrease in inhibition. Our results indicate that small increases in noise in either excitation or inhibition or both near a critical point may dampen firing rates. Thus, according to our computational studies, an increase in firing could result from a decline in inhibition. Such interesting aspects of network behavior will be pursued in a later article.

The differential equations for the approximate first- and second-order moments of the model were derived in section 3. The system consists of 6 equations for the means, 6 for the variances, and 15 for the covariances, giving 27 differential equations in total. These were solved numerically with a Runge-Kutta routine and the results compared with those obtained by simulation. When the level of excitation is too weak to evoke regular firing, the assumptions for the validity of the moment method are not satisfied and agreement with simulation is generally poor. However, when the firing rate was larger, either with stronger excitatory synaptic input or an additional added depolarizing current and when the noise was of small amplitude, then in a few examples, good agreement was found between the two methods over small time intervals, as in the Fitzhugh-Nagumo system with additive noise (Rodriguez & Tuckwell, 1996).

It is interesting to ask whether the solutions of the moment equations display any of the properties of the original system of stochastic differential equations with regard to the suppression of spiking by small noise near the critical input for repetitive spiking. It was found that the moment equations with no noise and , gave repetitive spiking with 12 spikes in 200 ms, as in the top right part of Figure 4. When small amplitude noise was added as in the records of Figure 4, spike numbers were reduced. With and no inhibitory noise, there were two spikes, but the second spike did not repolarize fully. With noise of twice the amplitude, there occurred only an early single complex spike. Similar solutions were obtained with further increases in . Surprisingly, noise amplitudes from 0.00001 to 0.0000001 all gave rise to 3 spikes. Steadily reducing the noise amplitude further eventually gave rise to the 12 spikes of the noise-free system. Thus, although small-amplitude noise did suppress the spiking and there was a decline in spike number with increasing noise, there was no evidence of a minimum. Similar observations were obtained with only inhibitory noise, but there were 2 spikes for and 1 complex spike for .

To further investigate the differences between the responses of system of SDEs and the moment equations some results for the two systems are shown in Figure 15. (For explanation, see the figure caption.) These results demonstrate that for the parameter values given in the previous paragraph, with excitatory noise level at the low level, , the dynamical properties of the two systems are very different, which underlies the fact that agreement between the simulation and moment equation results is found in only a few cases. In simpler systems such as the Fitzhugh-Nagumo model, where there are only two components and five moment equations, there is a wider agreement between the two sets of results (Rodriguez & Tuckwell, 1996). Perhaps it is relevant to point out that in the one-dimensional models, such as the integrate-and-fire or leaky integrate-and-fire model, the first- and second-order moments of the subthreshold membrane potential are most often calculable exactly and there are no covariances to consider.

Figure 15:

Comparison of voltage versus potassium activation variable trajectories for the system of six stochastic differential equations (left panel) with those of their means in the system of 27 deterministic differential equations for the first- and second-order moments (right panel). Here, there is excitation and inhibition near threshold for repetitive firing as in Figure 4. In both cases the blue curves, which are practically identical, are for the case of no noise, showing the approach to a limit cycle. (Left) One of the stochastic trajectories (red) stays close to the limit cycle, whereas the other (black) collapses to an equilibrium point around P near the rest point to terminate the spiking. (Right) The (deterministic) solution with noise (black curve) begins to oscillate on its second excursion close to the limit cycle and collapses to an equilibrium point near Q to terminate the spiking.

Figure 15:

Comparison of voltage versus potassium activation variable trajectories for the system of six stochastic differential equations (left panel) with those of their means in the system of 27 deterministic differential equations for the first- and second-order moments (right panel). Here, there is excitation and inhibition near threshold for repetitive firing as in Figure 4. In both cases the blue curves, which are practically identical, are for the case of no noise, showing the approach to a limit cycle. (Left) One of the stochastic trajectories (red) stays close to the limit cycle, whereas the other (black) collapses to an equilibrium point around P near the rest point to terminate the spiking. (Right) The (deterministic) solution with noise (black curve) begins to oscillate on its second excursion close to the limit cycle and collapses to an equilibrium point near Q to terminate the spiking.

Finally, we have seen that the inhibition of firing can be induced by noisy synaptic input of excitatory or inhibitory character or both, and that the phenomenon of inverse stochastic resonance may exist when there is noise in inhibitory inputs alone. The moment equations have been shown to exhibit dampening of firing rates with noise and agreement with simulations in some cases, but their complexity makes their ability to predict firing rates of neurons modeled in the way presented here somewhat limited.

Appendix:  HH Coefficients, Parameters, and Initial Values

A.1  Coefficients and Their First and Second Derivatives

Recalling that here V is depolarization and not membrane potential (cf. the original forms in HH (1952)), the coefficients in the auxiliary equations are the following standard ones:
formula
In the moment equations, we require their first- and second-order derivatives. The latter are
formula
The corresponding second derivatives are found to be
formula
where
formula

A.2  Standard Parameters and Initial Values

The values of the HH parameters employed in the model are the standard ones: , , , , , , and . The initial (resting) values are ,
formula
The numerical values of the last three quantities are 0.0529, 0.5961, and 0.3177, respectively.

Acknowledgments

We appreciate helpful correspondence with Alain Destexhe at CNRS, Gif-sur-Yvette, France. The work is part of the Dynamical Systems Interdisciplinary Network, University of Copenhagen. We thank the referees for raising many interesting points that have been addressed in the discussion.

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