Transition to complete synchronization via near-synchronization in two coupled chaotic neurons

The synchronization transition in two coupled chaotic Morris-Lecar （ML） neurons with gap junction is studied with the coupling strength increasing. The conditional Lyapunov exponents, along with the synchronization errors are calculated to diagnose synchronization of two coupled chaotic ML neurons. As a result, it is shown that the increase in the coupling strength leads to incoherence, then induces a transition process consisting of three different synchronization states in succession, namely, burst synchronization, near-synchronization and embedded burst synchronization, and achieves complete synchronization of two coupled neurons finally. These sequential transitions to synchronization reveal a new transition route from incoherence to complete synchronization in coupled systems with multi-time scales.

Complex transitions between spike, burst or chaos synchronization states in coupled neurons with coexisting bursting patterns

We investigated the synchronization dynamics of a coupled neuronal system composed of two identical Chay model neurons. The Chay model Showed coexisting period-1 and period-2 bursting patterns as a parameter and initial values are varied. We simulated multiple periodic and chaotic bursting patterns with non-（NS）, burst phase （BS）, spike phase （SS）, complete （CS）, and lag synchronization states. When the coexisting behavior is near period-2 bursting, the transitions of synchronization states of the coupled system follows very complex transitions that begins with transitions between BS and SS, moves to transitions between CS and SS, and to CS. Most initial values lead to the CS state of period-2 bursting while only a few lead to the CS state of period-I bursting. When the coexisting behavior is near period-1 bursting, the transitions begin with NS, move to transitions between SS and BS, to transitions between SS and CS, and then to CS. Most initial values lead to the CS state of period-1 bursting but a few lead to the CS state of period-2 bursting. The BS was identified as chaos synchronization. The patterns for NS and transitions between BS and SS are insensitive to initial values. The patterns for transitions between CS and SS and the CS state are sensitive to them. The number of spikes per burst of non-CS bursting increases with increasing coupling strength. These results not only reveal the initial value- and parameter- dependent synchronization transitions of coupled systems with coexisting behaviors, but also facilitate interpretation of various bursting patterns and synchronization transitions generated in the nervous system with weak coupling strength.

Standing Wave Solutions in Nonhomogeneous Delayed Synaptically Coupled Neuronal Networks

The authors establish the existence and stability of standing wave solutions of a nonlinear singularly perturbed system of integral differential equations and a non- linear scalar integral differential equation. It will be shown that there exist six standing wave solutions （（u（x,t）,w（x,t）） = （U（x）,W（x）） to the nonlinear singularly perturbed system of integral differential equations. Similarly, there exist six standing wave so- lutions u（x,t） = U（x） to the nonlinear scalar integral differential equation. The main idea to establish the stability is to construct Evans functions corresponding to several associated eigenvalue problems.

Mathematical modeling of the biphasic dopaminergic response to glucose

In this work, we specify potential elements of the brain to sense and regulate the energy metabolism of the organism. Our numerical investigations base on neurochemical experiments demonstrating a biphasic association between brain glucose level and neuronal activity. The dynamics of high and low affine KATP channels are most likely to play a decisive role in neuronal activity. We develop a coupled Hodgkin-Huxley model describing the interactive behavior of inhibitory GABAergic and excitatory dopaminergic neurons projecting into the caudate nucleus. The novelty in our approach is that we include the synaptic coupling of GABAergic and dopaminergic neurons as well as the interaction of high and low affine KATP channels. Both are crucial mechanisms described by kinetic models. Simulations demonstrate that our new model is coherent with neurochemical in vitro experiments. Even experimental interventions with glibenclamide and glucosamine are reproduced by our new model. Our results show that the considered dynamics of high and low affine KATP channels may be a driving force in energy sensing and global regulation of the energy metabolism, which supports central aspects of the new Selfish Brain Theory. Moreover, our simulations suggest that firing frequencies and patterns of GABAergic and dopaminergic neurons are correlated to their neurochemical outflow.