Geometrical analysis of codimension-1 bifurcations of rest state in two-dimensional neuronal models and the role of bifurcations in determining the class of neuronal excitability

Background and Aim : the intrinsic properties of nerve cells, along with the synaptic properties of connections between neurons and the topology of network connectivity play an important role in determining the neuronal population rhythms.neurons are considered as a nonlinear dynamical systems which look at the synaptic input through the prism of their own intrinsic dynamics. That is, they respond to input according to their inherent features, that this response may be in the form of subthreshold oscillations or repetitive spikes. In this paper we view neurons from the perspective of dynamical systems, and we use geometrical methods to illustrate codimension-1 bifurcations of rest stat in two-dimensional neuronal models. Also, we study the role of bifurcations in determining the neuro-computational properties, particularly neuronal excitability classes.Methods : here we investigate bifurcations of rest state occurred in two-dimensional neuronal models using geometric analysis of phase space, and also we refer to the essential role of the nullclines in locating the equilibria and in determining the shape of vector field in the neighborhood of stable equilibrium in the moment of bifurcation. In addition we use canonical models to illustrate the dynamics of class 1 excitable systems.Results : A strange situation regarding to class 1 excitable neurons occurs when the threshold manifold curves around the rest state and a strong inhibitory input can evoke spike. Also a class 2 excitable system near an Andronov-Hopf bifurcation possesses an substantial information processing capability. System s response to a pair (or a sequence) of stimuli depends on the timing between the stimuli. A pair of relatively strong pulsed perturbations (a doublet), may or may not evoke an action potential depending on its interspike interval. When the interval is close the period of the damped oscillations, then the effect of the perturbations can accumulate. On the other hand, when the interval is less or more than the period of the damped oscillations, afterward the perturbations may effectively cancel each other. Thus, the interspike interval in doublets plays an important role in eliciting response in postsynaptic neurons depending on their frequency. This provides a powerful mechanism for selective communication between such neurons. In particular, such neurons can multiplex; i.e. send many signals via a single transmission line.Conclusion : the bifurcations determine excitable properties of neurons, and hence their neuro-computational properties. Class 1 neural excitability corresponds to the resting state disappearing via saddle-node on invariant circle bifurcation, and action potentials can be generated with arbitrarily low frequency. In contrast, class 2 neural excitability corresponds to the rest state disappearing via saddle-node (off invariant circle) bifurcation or losing stability via subcritical or supercritical Andronove-Hopf bifurcation, which action potentials are generated in a certain frequency band.

کلیدواژه ها:

Neuronal model ، Andronove-Hppf bifurcation ، Saddle-node on invariant circle bifurcation ، Class 1 neural excitability ، Class 2 neural excitability