Representation of single neuron dynamics using 1-D and 2-D Discrete dynamical systems

Biological neurons are typically modeled using the Hodgkin-Huxley formalism, which requires significant computational power to simulate. However, since realistic neural network models require thousands of synaptically coupled neurons, a faster approach is needed. Discrete dynamical systems are promising alternatives to continuous models, as they can simulate neuron activity in far fewer steps. Many existing discrete models are based on Poincaré-map-like approaches, which trace periodic activity at a cross section of the cycle. However, this approach is limited to periodic solutions. Biological neurons have many key properties beyond periodicity, such as the minimum applied current required for a resting cell to generate an action potential. To address these properties, we propose a discrete dynamical system model of a biological neuron that incorporates the threshold dynamics of the Hodgkin-Huxley model, the logarithmic relationship between applied current and frequency, modifications to relaxation oscillators, and spike-frequency adaptation in response to modulatory hyperpolarizing currents. It is important to note that several critical parameters are transferred from the continuous model to our proposed discrete dynamical system. These parameters include the membrane capacitance, leak conductance, and maximum conductance values for sodium and potassium ion channels, which are essential for accurately simulating the behavior of biological neurons. By incorporating these parameters into our model, we can ensure that it closely approximates the continuous model's behavior, while also offering a more computationally efficient alternative for simulating neural networks.