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2.6 Collective Response of Neurons

The basis for mathematical modeling of neurons and their computational capabilities dwells on two considerations, namely, the associated threshold logic and the massive interconnections between them. The synergic or collective response of the neural complex is essentially stochastical due to the randomness of the interconnections and probabilistic character of individual neurons. A neural complex is evolved by the progressive multiplication of the interneuronal connections. As a result, the participation of an individual neuron in the collective response of the system becomes less strongly deterministic and more probabilistic. This gives rise to evoked responses of neurons being different each time to the repeated stimuli, though the reactions of the entire neuronal population (manifested as ECG, EEG, EMG, etc.) could be the same every time. Thus, the interconnected neuronal system with the stochastical input-output characteristics corresponds to a redundant system of parallel connections with a wide choice of ways for the signal propagation (or a von Neumann random switch — a computational basic hardware is realizable in terms of neural nets).

For the neurophysiological consideration that a neuron fires only if the total of the synapses which receive impulses in the period of latent summation exceeds the threshold, McCulloch and Pitts [7] suggested a highly simplified computational or logical neuron with the following attributes:

  A formal neuron (also known as the mathematical neuron, or logical neuron, or module) is an element with say m inputs (x1, x2, …, xm; m ≥ 1) and one output, O where m is an axonal output or a synaptic input of a neuron. Associating weights Wi (i ∈ m) for each input and setting threshold at VT, the module is presumed to operate at discrete time instants ti (i ∈ n). The module “fires” or renders an output at (n + 1)th instant along its axon, only if the total weight of the inputs simulated at time, n exceeds VT. Symbolically, O (n + 1) = 1 iff ςWixi(n) ≥ VT.
  The positive values of Wi (>0) correspond to excitatory synapses (that is, module inputs) whereas a negative weight Wi < 0 means that xi is an inhibitory input. McCulloch and Pitts showed that the network of formal neurons in principle can perform any imaginable computation, similar to a programmable, digital computer or its mathematical abstraction, namely, the Turing machine [30]. Such a network has an implicit program code built-in via the coupling matrix (Wi). The network performs the relevant computational process in parallel within each elementary unit (unlike the traditional computer wherein sequential steps of the program are executed).
  A neural net, or a modular net, is a collection of modules each operating in the same time-scale, interconnected by splitting the output of any module into a number of branches and connecting some or all of these to the inputs of the modules. An output therefore may lead to any number of inputs, but an input may come at most from one output.
  The threshold and weights of all neurons are invariant in time.
  The McCulloch-Pitts model is based on the following assumptions: Complete synchronization of all the neurons. That is, the activities of all the neurons are perceived in the same time-scale.
  Interaction between neurons (for example, the interactive electric fields across the neurons due to the associated impulses) is neglected.
  The influences of glial cell activity (if any) are ignored.
  Biochemical (hormonal, drug-induced) effects (on a short- or on long-term basis) in changing behavior of the neural complex are not considered.

In search of furthering the computational capabilities of a modular set, Hopfield in 1982 [31] queried “whether the ability of large collections of neurons to perform computational tasks may in part be a spontaneous collective consequence of having a large number of interacting simple neurons.” His question has the basis that interactions among a large number of elementary components in certain physical systems yield collective phenomena. That is, there are examples occurring in physics of certain unexpected properties that are entirely due to interaction; and large assemblies of atoms with a high degree of interaction have qualitatively different properties from similar assemblies with less interaction. An example is the phenomenon of ferromagnetism (see Appendix A) which arises due to the interaction between the spins of certain electrons in the atom making up a crystal.

The collective response of neurons can also be conceived as an interacting process and has a biological basis for this surmise. The membrane potential of each neuron could be altered by changes in the membrane potential of any or all of the neighboring neurons. After contemplating data on mammals involving the average separation between the centers of neighboring cell bodies and the diameters of such cell-bodies, and finding that in the cat the dendritic processes may extend as much as 500 μm away from the cell body, Cragg and Temperley [32] advocated in favor of there being a great intermingling of dendritic processes of different cells. Likewise, the axons of brain neurons also branch out extensively. Thus, an extreme extent of close-packing of cellular bodies can be generalized in the neuronal anatomy with an intermingling of physiological processes arising thereof becoming inevitable; and, hence, the corresponding neural interaction refers to a collective activity.


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