Control of the complexity of associative memory dynamics by neuronal adaptation

An abstract neural network model of the Hopfield type is extended to incorporate neuronal adaptation by defining the state of a neuron in terms of two variables, activity and excitability. The model is formulated to represent the regulation of the firing rate of action potentials in a biological system via the neuron cell membrane afterhyperpolarization by the effect of intracellular calcium ion concentration on the conductance of calcium sensitive potassium channels. It is shown that the complexity, and thus the exploratory degree, of associative memory dynamics are controlled by neuronal adaptability. At low adaptability, the dynamics have fixed point attractors corresponding to direct memory retrieval. In a subsequent region of adaptability values, a simple limit cycle persists with frequency increasing with adaptability. The range of frequencies agrees with that observed for theta rhythms of activity in the brain. A higher degree of freedom of the associative process corresponding to more complex dynamics, either limit cycles of varying complexity and period or chaotic behaviour, results at higher adaptability. In the brain, the neuronal adaptability is regulated by neuromodulators which suppress adaptation and increase absolute firing rates of action potentials. An associative process can be started at low concentration of neuromodulators as an exploratory search of state space during which firing rates are low. As the concentration of neuromodulators increases, the state space search becomes simpler cyclic and more restricted, and firing rates increase. Eventually, a particular stored state is retrieved and its activity is high. This correspondence between the complexity of associative memory dynamics and the concentration of neuromodulators is consistent with the observation for Alzheimer's disease of selective degeneracy of neurons releasing the neuromodulator acetylcholine. In an artificial neural network, inclusion of adaptation among neuronal properties allows control of the degree of freedom of associative processes and thus extends the range of possible applications.