![]() |
|
|||
![]() |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
![]() |
|
![]() |
[an error occurred while processing this directive]
The membrane potential of each neuron due to the interconnected configuration with the other cells could likewise be altered as a result of changes in membrane potential of any or all of the neighboring neurons. The closely packed neurons, as observed by Cragg and Temperley, hence permit the application of the theory of cooperative processes in which the cells are at two states of energy levels; and the interaction between the cells introduces a correlation between the states occupied by them. The whole assembly has a fixed amount of energy which is conserved but with no restriction to change from one configuration to the other. The interaction across the whole assembly also permits a proliferation of state changes through all the possible neuronal configurations. Each configuration, however, has a probability of occurrence; and hence the average properties of the whole assembly refer to the weighted averaging over all the possible configurations. In the mediating process by all-or-nothing impulses as mentioned before, the pertinent synaptic interaction could be either excitatory (hypopolarized) or inhibitory (hyperpolarized) interaction perceived as two different directional (ionic) current flows across the cellular membrane. Suppose the excitatory and inhibitory interactions are exactly balanced. Then the overall effect is a null interaction. If, on the other hand, the inhibitory process dominates, the situation is analogous to antiferromagnetism which arises whenever the atomic interaction tends to set the neighboring atomic magnets in opposite directions. Then, on a macroscopic scale no detectable spontaneous magnetism would prevail. In a neurological analogy, this corresponds to a zero potential difference across the cellular domains. It has been observed in neurophysiological studies that the hyperpolarization (inhibitory) process is relatively less prominent than the excitatory process; and the collective neuronal process would occur even due to an asymmetry of the order 1002:1000 in favor of the excitatory interactions. Cragg and Temperley hypothesized that a (large) set of M neurons analogously corresponds to a set of M atoms, each having spins ±1/2. A neuron is regarded as having two (dichotomous) states distinguished by the presence (all) or absence (none) of an action potential which may be correlated with the two independent states possible for an atom in a state having spin 1/2 with no further degeneracy* due to any other factors.
5.3 Concerns of GriffithAlmost a decade later, the qualitative analogy presented by Cragg and Temperley received a sharp criticism from Griffith [13,14] who observed that Cragg and Temperley had not defined the relation to ferromagnetic material in sufficient detail for one to know whether the analogies to macroscopic magnetic behavior should actually hold in the neural system. In the pertinent study, the neural assembly representing an aggregate of M cells with the dichotomous state of activity has 2M possible different states which could be identified by S = 1, , 2M associated with an M-dimensional hypercube; and Griffith observed that a superficial analogy here with the quantum statistical mechanics situation corresponds to a set of M subsystems, each having two possible quantum states as, for example, a set of M atoms each having a spin 1/2 (with no additional degeneracy due to any other possible contributing factors). Each of the 2M states has a definite successor in time so that the progress of the state-transitional process (or the neuronal wave motion) can be considered as a sequence i2 = z(i1) → i3 = z(i1) → ... and so on. Regarding this sequence, Griffith pointed out that in the terminal cycle of the state-transitional process, there are three possible situations, namely, a state of equilibrium with a probability distribution ρ(S0); and the other two are dichotomous states, identified as S1 ⇒ + SU and SU ⇒ - SL with the statistics depicted by the probability distributions ρ(S1) and ρ(S2), respectively. In computing the number of states which end up close to the equilibrium at the terminal cycle, Griffith indicated the following fundamental difference between the neural and the quantum situation: From the quantum mechanics point of view, the state-transitional probabilities (due to an underlying barrier potential φ) between two states S1 and S2 with probabilities ρ1,2 (and corresponding wave functions Ψ1,2) specified with reference to the equilibrium state, namely, S0 (with ρ0 ⇒ Ψ0) are equal in both directions. This is because they are proportional, respectively, to the two sides of the equation given by: The above relation is valid inasmuch as φ is Hermitian.* In the case of neural dynamics, however, Griffith observed that the possibilities of i2 = z(i1) and i1 = z(i2) are rather remote. That is, there would be no microscopic reversibility. There could only be a natural tendency for the microscopic parameter ρ1,2 to move near to ρo and there would not seem to be any very obvious reason for the successor function z to show any particular symmetry.
Copyright © CRC Press LLC
![]() |
![]() |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
![]() |
![]() |