# 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.

## Introduction

\label{1}

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.

## Methods

\label{2}

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

\label{2.1}

The LIF neuron is given by the equation

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

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