Evolution of neuron firing and connectivity in neuronal plasticity with application to Parkinson’s disease

We unify different modeling structures for neuron state evolution showing how they can be derived from an invariance requirement imposed to a dissipation inequality under diffeomorphism-based changes of observer in the physical space. This is a completely non-standard way of using the second law of thermodynamics, although in isothermal setting, a law otherwise commonly used for determining constitutive restrictions and admissibility conditions as those pertaining to shock waves. In this setting, we also consider a time-varying neuronal connectivity and derive the consistent structure of its evolution equation from the same invariance principle. In the case of Parkinson, the connectivity depends also on the distribution of calcium channels that bring dopamine excess. Under special conditions, we show how the connectivity probability distribution changes in time and is influenced by the one of calcium channels. We account for memory effects in the cortical matter, namely dependence on firing and connectivity histories.