Date Published: March 30, 2017
Publisher: Public Library of Science
Author(s): Mauricio Girardi-Schappo, Germano S. Bortolotto, Rafael V. Stenzinger, Jheniffer J. Gonsalves, Marcelo H. R. Tragtenberg, Dante R. Chialvo.
We introduce a new map-based neuron model derived from the dynamical perceptron family that has the best compromise between computational efficiency, analytical tractability, reduced parameter space and many dynamical behaviors. We calculate bifurcation and phase diagrams analytically and computationally that underpins a rich repertoire of autonomous and excitable dynamical behaviors. We report the existence of a new regime of cardiac spikes corresponding to nonchaotic aperiodic behavior. We compare the features of our model to standard neuron models currently available in the literature.
Modeling the brain is not a simple task. Usually scientists opt for simple models that provide insights about the original phenomenon one is trying to study. On the other hand, these models may lack some fundamental features that would play an important role in the considered phenomenon, specially when studying complex systems like the brain. Nowadays supercomputers have allowed the Neuroscience community to propose large-scale models for either brain functions or brain electrophysiology [1–5].
The KT is a two-dimensional model defined by Eqs (1a) and (1b) below. The KTz is a tridimentional model defined by Eqs (1a)–(1c). In both models the gain function f(u) is a hyperbolic tangent. To obtain the KTLog and KTzLog models we approximate the hyperbolic tangent by its first order Taylor expansion, the logistic function f(u) = u/(1 + |u|):
This section is dedicated to show that this map exhibits many dynamical behaviors of excitable systems, like neuronal cells and muscle cells, especially the action potential of cardiac cells .
Usually, the performance of a program depends on many uncontrollable variables, such as code style, language specific implementations, background memory operations and usage, function calls, concurrent processing due to operating system threads, etc. Thus, there is a diversity of measures for computational efficiency. Izhikevich  defines it through the amount of floating-point operations (FLOPs) the neuron model needs in order to evolve 1 ms (model time) of its dynamics, considering that a spike takes around 1 ms to rise and fall.
We studied a model of action potential generation using difference equations obtained through the first order Taylor approximation of the hyperbolic tangent KTz neuron model . We presented detailed bifurcation diagrams and fixed point stability diagrams for our new model, the so-called KTzLog neuron. We have shown that the KTzLog neuron reproduces many neuronal excitatory and autonomous behaviors observed experimentally. We have compared our model’s computational performance to other widely used models and concluded that its efficiency is comparable to that of the most efficient neuron models, both in isolation or in a network.