A SIMPLE MODEL AS A TESTBED FOR FUNDAMENTAL NEURONAL PROPERTIES
Sometimes a simple mathematical model suffices for both testing and demonstrating fundamental dynamical properties of complex systems. One such example is the logistic map, a one-dimensional iterative process capable of exhibiting very complicated behaviors, including the well-known period doubling cascade in transitions between periodic behavior and chaos. Here we extend this idea to the case of neurological processes, where complex neuronal model equations exhibit intriguingly complicated transitions. These transitions have been observed both in the single neuron and in networks of neurons where fundamental features of the single neuron is passed on to the collective of the network. The question then arises on whether these are properties associated only with the complex and more complete models, or are they also present, albeit perhaps hidden in the dynamics of simpler systems. We test the idea downgrading from the Huber-Brown model (a set of four differential equations embodied with elaborated functions for specific ion channels) to the Fitz-Nagumo equations (a set of three much simpler equations). This model is based off of the Bonhoeffer-van der Pol model where, in addition to three parameters, one variable is the membrane potential, another is a recovery variable, and a third is the magnitude of a stimulus current. This is in sharp contrast to the Huber-Brown model which contains 23 parameters in addition to four time-dependent variables. In this presentation we also provide some information about typical features of the FirzHugh-Nagumo model, including its “all-or-none” response associated with the existence of no threshold for firing and absence of the saddle equilibrium. We will also discuss briefly the electronic circuit implementation of the model, done originally by J.-I. Nagumo.
Patterson, Jennifer, "A SIMPLE MODEL AS A TESTBED FOR FUNDAMENTAL NEURONAL PROPERTIES" (2019). University Research Symposium. 334.