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Notwithstanding the fact that the aforesaid similarities do prevail between the neurons and the magnetic spins, major inconsistencies also, persist between these two systems regarding the synaptic coupling versus the spin interactions (Table 5.2). Mainly, the inconsistency between neurons with inherent asymmetric synaptic couplings and symmetric spin-glass interactions led Griffith [14] to declare the aggregate of neurons versus magnetic spin analogy as having “no practical value”. Nevertheless, several compromising suggestions have been proposed as discussed earlier showing the usefulness of the analogy (between the neurons and the magnetic spins).

Table 5.1 Ising Spin System versus Neuronal System: Analogical Aspects
Magnetic Spin System Neuronal System
Interacting magnetic spins represent a collective process. Interacting neurons represent a collective process.
Dichotomous magnetic spin states: ±Si. Dichotomous cellular potential states: σi = 0 or 1.
Exchange interactions are characterized by strengths of interaction. Synaptic couplings are characterized by weights of synaptic connections.
Competition between ferromagnetic and antiferromagnetic exchange interactions. Competition between the excitatory and inhibitory processes.
A set of M magnetic dipoles each with two spins (±1/2). A set of M neurons each with two potential states, 0 or 1.
Condition of metastability against single-spin flips. Cellular state-transition crossing a threshold (metastable) state.
Phase transition from paramagnetism to ferromagnetism at a critical temperature (Curie point). Onset of persistent firing patterns at a critical potential level.
A spin is flipped iff, the Hamiltonion (Lyapunov functional of energy) sets the dynamics of the spins to a ground-state. A state of a neuron is changed iff, the Hamiltonion sets the dynamics of the neurons to converge to a local minimum (ground-state).

Table 5.2 Ising Spin System versus Neuronal System: Contradictions and Inconsistencies
Magnetic Spin System Neuronal System
Microscopic reversibility pertaining to the magnetic spin interactions is inherent to the strength of coupling between the exchange interactions being symmetrical. Symmetric weighting of neuronal interaction is questionable from the physiological viewpoint. This implies the prevalence of unequalness between the number of excitatory and inhibitory synapses.
That is, in the magnetic spin exchange interactions, the coupling coefficients Jij = Jji. In the neuronal cycle of state-transitions, the interconnecting weights WijWji.
The physical (molecular) arrangement of magnetic dipoles facilitates the aforesaid symmetry. The physiological reality forbids the synaptic forward-backward symmetric coupling.
Symmetry in the state-transition matrix. Asymmetry in the state-transition matrix.
Diagonalizable transition matrix. Nondiagonalizable transition matrix.
No anisotropy in magnetic dipole orientations unless dictated by an external magnetic influence. Anisotropy is rather inherent leading to a persistent order (in time as depicted by Little or in space as discussed in Section 5.10).
Hamiltonians obey the principle of detailed balance. Only a subclass of Hamiltonians obey the principle of detailed balance.

The assumption of symmetry and the specific form of the synaptic coupling in a neuronal assembly define what is generally known as the Hopfield model. This model demonstrates the basic concepts and functioning of a neural network and serves as a starting point for a variety of models in which many of the underlying assumptions are relaxed to meet some of the requirements of real systems. For example, the question of Wij being not equal to Wji in a neural system was addressed in a proposal by Little (as detailed in the previous section), who defined a time-domain long-range order so that the corresponding anisotropy introduces bias terms in the Hamiltonian relation, making it asymmetric to match the neuronal Hamiltonian. That is, Little’s long-range order as referred to neurons corresponds to a time-domain based long-time correlation of the states; and these persistent states (in time) of a neuronal network are equated to the long-range (spatial) order in an Ising spin system.


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