LOGNORM: Conductance-based refractory density approach and its extension for lognormal input weight distribution

Conductance-based refractory density (CBRD) approach is an efficient approach that describes firing activity of a statistical ensemble of uncoupled Hodgkin-Huxley-like neurons each receiving individual gaussian noise and common time-varying deterministic input. However, the approach requires comparison with experiments and extension to the case of more realistic synaptic connectivity. Here we verify a CBRD model through comparison with experimental data and then generalize the model for the case of lognormal (LN) distribution of input weights within a population. We show that the model with equal weights efficiently reproduces post-spike time histogram and membrane voltage of experimental multiple-trial response of single neurons to step-wise current injection and reveals much more rapid reaction of firing-rate than voltage. Slow adaptive potassium channels strongly affect the shape of the responses. Next, we derive a computationally efficient CBRD model for a population with the LN input weight distribution and compare it to the original model with equal input weights. The analysis shows that the LN distribution does: (i) provide faster response; (ii) eliminate oscillations; (iii) lead to higher sensitivity to weak stimuli; (iv) increase the coefficient of variation of interspike intervals. In addition, a simplified, firing-rate type model is tested, showing better precision in the case of LN distribution of weights. More generally, the CBRD approach can be recommended for complex, biophysically detailed simulations of interacting neuronal populations, and the modified firing-rate type model for computationally reduced simulations.



Known neuronal population models belong either to the type of very simplified models such as firing-rate (FR) or neural mass models or to those that are based on probability-density approach (PDA). FR-models are usually expressed by algebraic or ordinary differential equations and easy to analyze. However, they are not fully adequate for transient process simulations because of their basic assumption that neurons are always desynchronized. A probability density approach avoids this assumption, however this approach is commonly applied to simplified, one-variable neurons like linear and nonlinear integrate-and-fire, or spike-response model or so (Gerstner 2002), (Eggert 2001), (Knight 2000), (Naud 2012), (Nicolas Fourcaud-Trocmé 2003). This limitation has been overcome by a conductance-based refractory density approach (CBRD) that is applicable to regular-spiking, adaptive and fast-spiking neurons described in terms of Hodgkin-Huxley-like approximations (Chizhov 2007). The CBRD approach has been proposed for an infinite set of such uncoupled neurons receiving a common input and an individual gaussian white noise-current. The model was later extended to the case of color noise (Chizhov 2008). Such model is found to be quite efficient for simulation of coupled populations (Chizhov 2013). The approach is worth being further developed and used in analysis of experimental data. In the present paper, the CBRD model is applied to experimental data revealing rapid population response to a weak input. Matching the model to experimental dataset validates the model, which then reveals the influence of neuronal parameters on the response. Further, the model is generalized for the case of lognormal distribution of the weights of the input received by different neurons of a population, which has many physiological implications (Teramae 2014). In addition, a simplified, firing-rate type model is tested to simulate a population with lognormal weights.



For the purpose of clarifying the ideas of the CBRD-approach and its extension to the lognormal input weight distribution, we present in this section the models for simplified, integrate-and-fire neurons. The details of the model for adaptive Hogkin-Huxley like neurons can be found in (Chizhov 2007) or (Chizhov 2013).

Leaky integrate-and-fire neuron with noise


The LIF neuron is given by the equation

\begin{equation} \label{e221} \label{e221}C~{}{dV\over dt}=-(g_{L}+s(t))(V-V_{rest})+I(t)+\sigma~{}\xi(t),\\ \end{equation}

where \(\xi(t)\) is a gaussian white noise characterized by its mean value, \(<\xi(t)>=0\), and auto-correlation \(<\xi(t)\xi(t^{\prime})>=C/g_{L}~{}\delta(t-t^{\prime})\); \(\sigma\) is the noise amplitude. The neuron fires when the potential \(V\) crosses the threshold \({V_{th}}\). Immediately after the spike the potential \(V\) is reset to \(V_{reset}\). The LIF neuron is characterized by the capacitance \(C\) and the leak conductance \(g_{L}\). The input is determined by two signals, the synaptic current \(I(t)\) that is measured at the voltage level equal to \(V_{rest}\), and the total synaptic conductance \(s(t)\). The effective membrane time constant is \(\tau_{m}=C/(g_{L}+s(t))\).

We will refer to as a population an infinite number of eq.(\ref{e221})-based LIF neurons receiving a deterministic 2-d input \((I(t),s(t))\) and individual for each neuron noise. The population firing rate is defined as a sum of all spikes, \(n^{act}\), from neurons of the population over a short time window \(\Delta t\), divided by the number of neurons, \(N\). After taking the limits of \(N\rightarrow\infty\) and \(\Delta t\rightarrow 0\), the firing rate \(\nu\) is obtained as

\begin{equation} \label{e2013} \label{e2013}\nu(t)=\lim_{\Delta t\rightarrow 0}\lim_{N\rightarrow\infty}\frac{1}{\Delta t}~{}\frac{n^{act}(t;~{}t+\Delta t)}{N}.\\ \end{equation}

The steady-state firing rate is given by the solution from (Johannesma 1968)

\begin{aligned} \label{e223} \label{e223}\nu^{SS} & = & \Biggl{(}\frac{C}{g_{L}+s(t)}~{}\sqrt{\pi}\int_{(V_{reset}-U)/\sigma_{V}\sqrt{(}2)}^{({V_{th}}-U)/\sigma_{V}\sqrt{(}2)}\exp(u^{2})(1+\hbox{erf}(u))~{}du\Biggr{)}^{-1}, \\ \label{e224}U & = & V_{rest}+I/(g_{L}+S), \\ \label{e225}\sigma_{V} & = & \frac{\sigma}{\sqrt{2}(g_{L}+s)}\\ \end{aligned}

where \(\sigma_{V}\) is the voltage dispersion; here \(U\) is the depolarization potential in the steady-state.

Conductance-based refractory density (CBRD) approach for a population of integrate-and-fire neurons


As justified in (Chizhov 2007), (Chizhov 2008), the firing rate for a population of LIF neurons is well approximated by the system of equations for the refractory density \(\rho(t,t^{*})\) and the averaged across noise realizations membrane potential \(U(t,t^{*})\), where \(t^{*}\) is the time elapsed since the last spike. Thereby, the CBRD method distinguishes neurons only according to their \(t^{*}\)-variable. In order words, the state of each neuron is parameterized by this phase variable. For the particular case of a LIF-neuron, its only state variable is \(V\), whereas for more complex neuron models the other state variables, the gating variables of ionic channels, are also parameterized by \(t^{*}\), which is the reduction to an only one phase variable description that rather precisely saves the information about neuronal states because of the same history of input for all neurons (Chizhov 2007). Returning to LIF-neurons, the equations for \(\rho(t,t^{*})\) and \(U(t,t^{*})\) are as follows

\begin{aligned} \label{e241} \label{e241}\frac{\partial\rho}{\partial t}+\frac{\partial\rho}{\partial t^{*}} & = & -\rho~{}H(U), \\ \label{e242}C\Biggl{(}\frac{\partial U}{\partial t}+\frac{\partial U}{\partial t^{*}}\Biggr{)} & = & -(g_{L}+s(t))(U-V_{rest})+I(t),\\ \end{aligned}

where \(H\) is a hazard function which is defined below. The boundary conditions are

\begin{aligned} \label{e2425} \label{e2425}\nu(t)\equiv\rho(t,0)=\int\limits_{+0}^{\infty}\rho Hdt^{*}\\ \end{aligned}

and \(U(t,0)=V_{reset}\), where \(\nu(t)\) is the population firing rate. When calculating the dynamics of a neuronal population, the integration of eq.(\ref{e242}) determines the distribution of not-noisy voltage \(U\) across \(t^{*}\). The effect of crossing the threshold and the diffusion due to noise are evaluated by the \(H\)-function and affect the equation for \(\rho\). The result of the integration of eq.(\ref{e241}) is the distribution of \(\rho\) across \(t^{*}\) and the firing rate \(\nu\) calculated from (\ref{e2425}).

The hazard function \(H\) is defined as the probability for a single neuron to generate a spike, if known actual neuron state variables. The hazard function \(H\) has been approximated in (Chizhov 2007) for the case of white noise and in (Chizhov 2008) for the case of color noise as a function of \(U(t)\) and \(s(t)\), and parameters \(\sigma\), \({V_{th}}\) and the ratio of membrane to noise time constants \(k=\tau_{m}/\tau_{Noise}\):

\begin{aligned} \label{e243} \label{e243}H & (U) & =A+B, \\ A & = & \frac{1}{\tau_{m}}e^{0.0061-1.12~{}T-0.257~{}T^{2}-0.072~{}T^{3}-0.0117~{}T^{4}}\biggl{(}1-(1+k)^{-0.71+0.0825(T+3)}\biggr{)},\nonumber \\ B & = & -\sqrt{2}~{}\biggl{[}{dT\over dt}\biggr{]}_{+}\tilde{F}(T),~{}~{}~{}\tilde{F}(T)=\sqrt{2\over\pi}~{}{\exp(-T^{2})\over 1+\hbox{erf}(T)},~{}~{}~{}T=\frac{{V_{th}}-U}{\sqrt{2}~{}\sigma_{V}},\nonumber \\ \end{aligned}

where \(T\) is the membrane potential relative to the threshold, scaled by noise amplitude; \(A\) is the hazard for a neuron to cross the threshold because of noise, derived analytically in (Chizhov 2007) and approximated by exponential and polynomial for convenience; \(B\) is the hazard for a neuron to fire because of depolarization due to deterministic drive, i.e. the hazard due to drift in the voltage phase space. Note that the \(H\)-function is independent of the basic neuron model and does not contain any free parameters or functions for fitting to any particular case. In this aspect, the \(H\)-function can be useful not only as a component of a population model, but as well for analysis tasks such as evaluation of neuronal susceptibility (Payeur 2015), effects of noise etc.

Firing-rate model for integrate-and-fire neurons


Conventional firing-rate type models often fail to describe well a population activity in transient states. A modified firing-rate model has been proposed in (Chizhov 2007a), which improves the transient solutions to step-like or complex-shape inputs (Buchin 2010). Below such model is given for the LIF-neuron population.

The subthreshold voltage \(U(t)\) is calculated as

\begin{equation} \label{e231} \label{e231}C~{}{dU\over dt}=-(g_{L}+s(t))(U-V_{rest})+I(t),\\ \end{equation}

and the firing rate is calculated by the following formula

\begin{aligned} \label{e232} \label{e232}\nu & = & \bigl{[}\nu^{SS}+\nu^{US}\bigr{]}_{+}, \\ \label{e234}\nu^{US} & = & {1\over\sqrt{2\pi}\sigma_{V}}{dU\over dt}\exp\Biggl{(}-{({V_{th}}-U)^{2}\over 2\sigma_{V}^{2}}\Biggr{)},\\ \end{aligned}

where the function \([x]_{+}\) is defined as \(x\) for \(x>0\) and \(0\) otherwise. The term \(\nu^{SS}\) is the steady-state solution (\ref{e223}). The essence of the model modification consists in the term \(\nu^{US}\), which evaluates the firing rate in response to fast excitation in assumption of a frozen gaussian distribution of neuronal membrane potentials crossing the threshold.



Validation of CBRD-model: comparison with analytical and Monte-Carlo solutions


To validate the CBRD-model based on eqs.(\ref{e221}-\ref{e223}) we compare it with Monte-Carlo (MC) simulation on a test problem of step-wise current stimulation. In MC simulations eq.(\ref{e221}) has been integrated 4000 times with different realizations of noise. The firing rate has been calculated from eq.(\ref{e2013}). As seen in Fig.\ref{F-1}A, the CBRD and MC solutions are close; they converge with the increase of the number of neurons in MS simulations.

For LIF neuron with white noise in steady-state, the analytical solution for the firing rate is known, it is given by eq.(\ref{e223}-\ref{e225}). The CBRD-model matches to this solution with a high precision, as demonstrated by Fig.\ref{F-1}B. Important, that the both solutions converge not only for variable input current but also for the input conductance.

Comparison of CBRD-model with experimental data


According to above introduced definition of a population, the neurons are independent from each other, thus, such population is equivalent to a number of noise realizations for a single neuron. Hence, statistics of the behavior of a neuron receiving both deterministic and stochastic inputs is governed by the same model as a single population. That is why, we compare the CBRD-model with experimental multiple registrations in a single neuron (Tchumatchenko 2011). A weak step-wise current was injected into the neuron. The statistics of spiking response is characterized by the post-stimulus spike-time histogram (PSTH) which corresponds to the firing rate for a population. The model has been taken from (Chizhov 2007). In this model a basic single neuron had a few potassium ionic channels. The adaptation was taken into account in the forms of the M-current and the current of afterhyperpolarization (AHP-current) which effectively approximates the effects of potassium calcium-dependent currents. The spike trains in response to step-wise current injection reveals the similarity of the modeled and registered neuronal responses (compare Figure 1A,D in (Tchumatchenko 2011)) with Fig. \ref{Fig1ABCD}A,C) and the effects of the adaptation. As to population response, the experimental PSTH (Figure 1E in (Tchumatchenko 2011)) is similar to the modeled firing rate (Fig. \ref{Fig1ABCD}D). The response in the model is as fast as in the experiment, the timing of peaks, the amplitudes of the firing rate and the character of relaxation to the steady-state are similar. Also we find out similar rapid reaction to the change of the amplitude of noise (Fig. \ref{Fig3B}).

The most affective factors determining the shape of firing response are the level of background activity before the stimulus and the slow potassium channels (Fig. \ref{Fig_SpontActivity_and_WhiteColorNoise}). As seen from comparison of Fig. \ref{Fig_SpontActivity_and_WhiteColorNoise}C and D with Fig. \ref{Fig_SpontActivity_and_WhiteColorNoise}B, the presence of spiking activity before the stimulus is critical for the rapidness of the response. The bigger the firing rate before stimulation the faster the response. This fact is consistent with the findings of the experimental study (Tchumatchenko 2011). Blockage of M- and A