Firing statistics and correlations in spiking neurons: A level-crossing approach

Abstract

We present a time-dependent level-crossing theory for linear dynamical systems perturbed by colored Gaussian noise. We apply these results to approximate the firing statistics of conductancebased integrate-and-fire neurons receiving excitatory and inhibitory Poissonian inputs. Analytical expressions are obtained for three key quantities characterizing the neuronal response to time-varying inputs: the mean firing rate, the linear response to sinusoidally-modulated inputs, and the pairwise spike-correlation for neurons receiving correlated inputs. The theory yields tractable results that are shown to accurately match numerical simulations, and provides useful tools for the analysis of interconnected neuronal populations.PACS numbers: 05.10. Gg,87.19.lj,87.19.ll,87.19.lm Understanding the dynamics of interconnected networks of neurons is of fundamental importance in theoretical neuroscience. An essential step in solving this problem is to determine the input-output relationship of individual neurons given an underlying biophysical model. For white-noise-driven integrate-and-fire (IF) neurons this problem is generally tractable (e.g., [1][2][3][4]), but efforts to integrate key aspects of neuronal signaling into the IF formalism have added to its complexity. First, synaptic input consists of discrete action potentials, which results in non-Gaussian voltage distributions and affects firing statistics [5][6][7][8]. Second, synaptic communication is mediated by two separate (excitatory and inhibitory) systems with distinct kinetics. The majority of theoretical studies include only one synaptic type and assume either fast (e.g., [9-11]) or slow [11,12] kinetics compared with the membrane integration time, but experimental evidence suggests that these timescales are often comparable [13]. Finally, neurons sharing presynaptic partners exhibit correlations in their synaptic input. While network models typically assume sparse connectivity for which correlations are negligible, recent reports suggest important functional roles for this type of "noise" correlation [14,15]. For IF neurons, obtaining the input-output relationship essentially involves computing moments of the first-passage-time (FPT) to threshold, but analytical solutions are rarely possible. Here, we show that for membrane-to-synaptic time constant ratios of the order of unity, level-crossing statistics provide good approximations to the FPT while retaining sufficient tractability to incorporate additional biological detail. Analytical expressions are given for the mean firing rate, pairwise spike-correlation and linear response to sinusoidally-modulated inputs, which compare favorably with numerical simulations of conductance-based IF neurons. The method thus provides a complete set of input- * Present address: Laboratory for Circuit Mechanisms of Sensory Perception, RIKEN Brain Science Institute, Wako, Saitama, Japan output properties needed for an analysis at the network level. Although the focus of the present paper is on neuroscience applications, the ...

Low-rate firing limit for neurons with axon, soma and dendrites driven by spatially distributed stochastic synapses

“…Compared with the one-dendrite model, we see from Figs 3b and 3d that the firing 237 rate for the two-dendrite model is significantly lower in the subthreshold regime but 238 converges to the same value when µ goes above threshold. This illustrates that even 239 simple differences in morphology affect stochastic and deterministic firing very However, despite the independence of λ, the firing-rate profile for this toy model is 245 distinct to that for the point-like leaky integrate-and-fire model, for which the second [29]. This indicates that spatial 247 structure by itself decreases the variance while increases derivative variance by a factor 248 1…”

“…One approximate analytical methodology, applicable to the low-firing-rate 30 limit driven by filtered synapses is the level-crossing method of Rice [27]. In this 31 approach, which has already been applied to compact neurons [28,29], a system without 32 post-spike reset is considered with the rate that the threshold is crossed from below 33 June 7, 2019 2/28 treated as a proxy for the firing rate; the upcrossing rate and firing rate for a system 34 with reset will be similar when the rate is low.…”

“…Firing rate approximated by the upcrossing rate 178 Full analytical solution of the partial differential Eqs (17,18) when coupled to the 179 integrate-and-fire mechanism does not appear trivial, even for the simple closed dendrite 180 model. However, a level-crossing approach developed by Rice [27] and exploited in many 181 other areas of physics and engineering, such as wireless communication channels [55], sea 182 waves [56], superfluids [57] and grown-surface roughness [58] has previously been applied 183 successfully to compact neuron models [28,29] and can be extended to spatial models. 184 The method provides an approximation for the mean first-passage time for any Gaussian 185 process in which the mean v , standard deviation σ v , and rate-of-change standard 186 deviation σv are calculable.…”

“…The approach can also be extended to 408 examine the dynamic firing-rate response to weakly modulated drive. This has already 409 been done for point-neurons using the upcrossing method [29,71,72] and would only 410 necessitate calculating the linear-response of voltage moments in the non-threshold case. 411 In summary, the extension of the upcrossing approach to spatially structured neuron 412 models provides an analytical in-road for future studies of the firing properties of 413 extended neuron models driven by spatio-temporal stochastic synaptic drive.…”

Low-rate firing limit for neurons with axon, soma and dendrites driven by spatially distributed stochastic synapses

“…One approximate analytical methodology, applicable to the low-firing-rate limit driven by filtered synapses is the level-crossing method of Rice [27]. In this approach, which has already been applied to compact neurons [28,29], a system without post-spike reset is considered with the rate that the threshold is crossed from below treated as a proxy for the firing rate. The upcrossing rate and firing rate for a model with an integrateand-fire mechanism will be similar when the rate is low, such that the effect of the previous reset has faded into insignificance by the time of the next spike.…”