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1.2 Stochastical Aspects and Physics of Neural Activity
The physics of neuronal activity, the proliferation of communication across the interconnected neurons, the mathematical modeling of neuronal assembly, and the physioanatomical aspects of neurocellular parts have been the topics of inquisitive research and in-depth studies over the past few decades. The cohesiveness of biological and physical attributions of neurons has been considered in the underlying research to elucidate a meaningful model that portrays not only the mechanism of physiochemical activities in the neurons, but also the information-theoretic aspects of neuronal communication. With the advent of developments such as the electron microscope, microelectrodes, and other signal-processing strategies, it has been facilitated in modern times to study in detail the infrastructure of neurons and the associated (metabolic) physiochemical activities manifesting as measurable electrical signals which proliferate across the interconnected neural assembly.
The dynamics of neural activity and communication/signal-flow considerations together with the associated memory attributions have led to the emergence of so-called artificial neurons and development of neural networks in the art of computational methods.
Whether it is the real neuron or its artificial version, the basis of its behavior has been depicted mathematically on a core-criterion that the neurons (real or artificial) represent a system of interconnected units embodied in a random fashion. Therefore, the associated characterizations depict stochastical variates in the sample-space of neural assembly. That is, the neural network depicts inherently a set of implemented local constraints as connection strengths in a stochastical network. The stochastical attributes in a biological neural complex also stem from the fact that neurons may sometimes spontaneously become active without external stimulus or if the synaptic excitation does not exceed the activation threshold. This phenomenon is not just a thermal effect, but may be due to random emission of neurotransmitters at the synapses.
Further, the activities of such interconnected units closely resemble similar physical entities such as atoms and molecules in condensed matter. Therefore, it has been a natural choice to model neurons as if emulating the characteristics analogous to those of interacting atoms and/or molecules; and several researchers have hence logically pursued the statistical mechanics considerations in predicting the neurocellular statistics. Such studies broadly refer to the stochastical aspects of the collective response and the statistically unified activities of neurons viewed in the perspectives of different algorithmic models; each time it has been attempted to present certain newer considerations in such modeling strategies, refining the existing heuristics and portraying better insights into the collective activities via appropriate stochastical descriptions of the neuronal activity.
The subject of stochastical attributions to neuronal sample-space has been researched historically in two perspectives, namely, characterizing the response of a single (isolated) neuron and analyzing the behavior of a set of interconnected neurons. The central theme of research that has been pursued in depicting the single neuron in a statistical framework refers to the characteristics of spike generation (such as interspike interval distribution) in neurons. Significantly, relevant studies enclave the topics on temporal firing patterns analyzed in terms of stochastical system considerations such as random walk theory. For example, Gerstein and Mandelbrot [2] applied the random walk models for the spike activity of a single neuron; and modal analysis of renewal models for the spontaneous single neuron discharges were advocated by Feinberg and Hochman [3]. Further considered in the literature are the markovian attributes of the spike trains [4] and the application of time-series process and power spectral analysis to neuronal spikes [5]. Pertinent to the complexity of neural activity, accurate modeling of a single neuron stochastics has not, however, emerged yet; and continued efforts are still on the floor of research in this intriguing area despite of a number of interesting publications which have surfaced to date. The vast and scattered literature on stochastic models of spontaneous activity in single neurons has been fairly comprised as a set of lecture notes by Sampath and Srinivasan [6].
The statistics of all-or-none (dichotomous) firing characteristics of a single neuron have been studied as logical random bistable considerations. McCulloch and Pitts in 1943 [7] pointed out an interesting isomorphism between the input-output relations of idealized (two-state) neurons and the truth functions of symbolic logic. Relevant analytical aspects have also since then been used profusely in the stochastical considerations of interconnected networks.
While the stochastical perspectives of an isolated neuron formed a class of research by itself, the randomly connected networks containing an arbitrary number of neurons have been studied as a distinct class of scientific investigations with the main objective of elucidating information flow across the neuronal assembly. Hence, the randomness or the entropical aspects of activities in the interconnected neurons and the self-re-exciting firing activities emulating the memory aspects of the neuronal assembly have provided a scope to consider the neuronal communication as prospective research avenues [8]; and to date the information-theoretic memory considerations and, more broadly, the neural computation analogy have stemmed as the bases for a comprehensive and expanding horizon for an intense research. In all these approaches there is, however, one common denominator, namely, the stochastical attributes with probabilistic considerations forming the basis for any meaningful analytical modeling and mathematical depictions of neuronal dynamics. That is, the global electrical activity in the neuron (or in the interconnected neurons) is considered essentially as a stochastical process.
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