Spiking Statistics of Excitatory Neuron with Feedback

Spiking Statistics of Excitatory Neuron with Feedback

Alexander Vidybida (Bogolyubov Institute for Theoretical Physics, Kyiv, Ukraine) and Kseniia Kravchuk (Bogolyubov Institute for Theoretical Physics, Kyiv, Ukraine)
DOI: 10.4018/joci.2012040101
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Abstract

Firing statistics of excitatory binding neuron (BN) is considered. The neuron is driven externally by a Poisson stream. Influence of feedback, which conveys every output impulse to the input with time delay , on the statistics of output spikes is studied. The resulting output stream is not Poissonian, and the authors obtain its inter-spike intervals (ISI) distribution for the case of BN, BN with instantaneous, , and delayed, , feedback. Output statistics of neuron with delayed feedback differs essentially from that found for the case of no feedback as well as from the case of instantaneous feedback. ISI distributions, found for delayed feedback, are characterized with jumps, derivative discontinuities and include -function type singularity. Also, for non-zero refractory time, the authors obtain multiple-ISI conditional probability density and prove, that delayed feedback presence results in non-Markovian statistics of neuronal firing. It is concluded, that delayed feedback presence can radically change neuronal firing statistics.
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Introduction

The role of input spikes timing in functioning of either single neuron, or neural net has been addressed many times, as it constitutes one of the main problem in neural coding. The role of timing was observed in processes of perception (MacLeod et al., 1988), memory (Hebb, 1949), objects binding and/or segmentation (Eckhorn, 1988). At the same time, where does the timing come from initially? In reality, some timing can be inherited from the external world during primary sensory reception. In auditory system, this happens for the evident reason that the physical signal, the air pressure time course, itself has pronounced temporal structure in the millisecond time scale, which is retained to a great extent in the inner hair cells output (Cariani, 2001). In olfaction, the physical signal is produced by means of adsorption-desorption of odor molecules, which is driven by Brownian motion. In this case, the primary sensory signal can be represented as Poisson stream, thus not having any remarkable temporal structure. Nevertheless, temporal structure can appear in the output of a neuron fed by a structureless signal. After primary reception, the output of corresponding receptor cells is further processed in primary sensory pathways, and then in higher brain areas. During this processing, statistics of poststimulus spiking activity undergoes substantial transformations, see, e.g. (Eggermont, 1991). After these transformations, the eventual pattern of activity is far away from the initial one. This process is closely related to the information condensation (König & Krüger, 2006). Such transformations also include the appearance of correlations in spiking activity of a given neuron in different moments of time (Levine, 1980; Lowen & Teich, 1992; Ratnam & Nelson, 2000; Nawrot et al., 2007; Farkhooi et al., 2009).

We now put a question: What kind of physical mechanisms might underlie these transformations? It seems that, among others, the following features are responsible for spiking statistics of a neuron in a network: (i) several input spikes are necessary for a neuron from a higher brain area to fire an output spike (see, e.g. Andersen et al., 1990); (ii) a neural net has numerous interconnections, which bring about feedback and reverberating dynamics in the net. Due to (i) a neuron must integrate over a time interval in order to gather enough input impulses to fire. As a result, in contrast to Poisson stream, the shortest ISIs between output spikes will no longer be the most probable. Due to reverberation, an individual neuron's output impulses can have some delayed influence on the input of that same neuron. This can be the source of positive feedback which results in establishing of dynamics partially independent of the stimulating input (compare with König & Krüger, 2006), and which governs neuronal spiking statistics. Due to the feedback, the future spiking moments of neurons will depend on present position of impulses in interconnections, which in turn depends on previous spiking moments of the same neurons. This may result in non-renewal and even non-Markovian statistics of neuronal firing (Kravchuk & Vidybida, 2013; Vidybida & Kravchuk, 2012).

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