Geometry for a brain. Optimal control in a network of adaptive memristors

Abstract

In the brain the relations between free neurons and the conditioned ones establish the constraints for the informational neural processes. These constraints reflect the systemenvironment state, i.e. the dynamics of homeocognit ive activities. The constraints allow us to define the cost function in the phase space of f ree neurons so as to trace the trajectories of the possible configurations at minimal cost whil e respecting the constraints imposed. Since the space of the free states is a manifold or a non orthogonal space , the minimum distance is not a straight line but a geodesic. The minimum condition is expressed by a set of ordinary differential equation ( ODE ) that in g eneral are not linear. In the brain there is not an algorithm or a physical field that regulates the computation, then we must consider an emergent process coming out of the neural collec tive behavior triggered by synaptic variability. We define the neural computation as the study of th e classes of trajectories on a manifold geometry defined under suitable constrain ts. The cost function supervises pseudo equilibrium thermodynamics effects that manage the computational activities from beginning to end and realizes an optimal control th rough constraints and geodetics. The task of this work is to establish a connection betw een the geometry of neural computation and cost functions. To illustrate the essential mathematical aspects we will use as toy model a Network Resistor with Adaptive Memory (Memristors).The information geometry here defined is an analog computation, therefore it does not suffer the limits of the Turing computation and it seems to respond to t he demand for a greater biological plausibility. The model of brain optimal control proposed here ca n be a good foundation for implementing the concept of intentionality,