Geometric Image of Neurodynamics

Abstract

We know that the brain is composed of simple neural units given by dendrites, soma, and axons. Every neural unit can be modelled by electrical circuits with capacitors and adaptive resistors. To study the neural dynamic we use special Ordinary Differential Equations (ODE) whose solutions give us the behaviour or trajectory of the neural states in time. The problem with ODE is in the definition of the parameters and in the complexity of the solutions that in many cases cannot be found. The key elements that we use are the multidimensional vector spaces of the electrical charges, currents and voltages. So currents and voltages are geometric references for states in the central neural system (CNS). Any neuro-biological architecture can be modelled by an adaptive electrical circuit or neuromorphic network that relates voltage with current by conductance matrix or on the contrary by impedance matrix. Given a straight line with a change of reference we reshape the straight line in a geodetic and in a new form for the distance. The change of the reference transforms a set of variables into another so this transformation is similar to a statement in the digital computer that we associate to the software. Every change of variables can be reproduced by a similar change of voltages (currents) into currents (voltages) by conductance (impedance) matrix. We use the CNS as a material support or hardware in the digital computer to realise the wanted transformation. In conclusion geometry fuses the digital computer structure with neuromorphic computing to give efficient computation where conceptual intention is the change of the reference space , while material intention is given by the neurodynamical processes modelled by the change of the electrical charge space where we define the metric geometry and distance.