Dynamics of period-doubling bifurcation to chaos in the spontaneous neural firing patterns

Abstract

Period-doubling bifurcation to chaos were discovered in spontaneous firings of Onchidium pacemaker neurons. In this paper, we provide three cases of bifurcation processes related to period-doubling bifurcation cascades to chaos observed in the spontaneous firing patterns recorded from an injured site of rat sciatic nerve as a pacemaker. Period-doubling bifurcation cascades to period-4 (π(2,2)) firstly, and then to chaos, at last to a periodicity, which can be period-5, period-4 (π(4)) and period-3, respectively, in different pacemakers. The three bifurcation processes are labeled as case I, II and III, respectively, manifesting procedures different to those of period-adding bifurcation. Higher-dimensional unstable periodic orbits (UPOs) can be detected in the chaos, built close relationships to the periodic firing patterns. Case III bifurcation process is similar to that discovered in the Onchidium pacemaker neurons and simulated in theoretical model-Chay model. The extra-large Feigenbaum constant manifesting in the period-doubling bifurcation process, induced by quasi-discontinuous characteristics exhibited in the first return maps of both ISI series and slow variable of Chay model, shows that higher-dimensional periodic behaviors appeared difficult within the period-doubling bifurcation cascades. The results not only provide examples of period-doubling bifurcation to chaos and chaos with higher-dimensional UPOs, but also reveal the dynamical features of the period-doubling bifurcation cascades to chaos.

Difference Between Intermittent Chaotic Bursting and Spiking of Neural Firing Patterns

“…The dynamics of chaotic behaviors in biology, especially in neuroscience, have been investigated for 30 years [Hayashi et al, 1982;Hayashi et al, 1983;Hayashi et al, 1996;Aihara et al, 1984;Aihara et al, 1985], attracting more and more attention. Chaotic neural firing patterns have been widely observed in biological experiments [Hayashi et al, 1982;Hayashi et al, 1983;Hayashi et al, 1996;Aihara et al, 1984;Aihara et al, 1985;Elbert et al, 1994;Xu et al, 1997;Di Mascio et al, 1999;Braun et al, 1999;Lovejoy et al, 2001;Gu et al, 2004;Gu, 2013b;Kanno et al, 2007;Jia et al, 2012]. The roles of chaos in the nervous system, such as neural coding, the processing of neuronal information, perception, memory, and higher brain functions, are also discussed [Rabinovich & Abarbanel, 1998;Schweighofer et al, 2004].…”

Dynamics of in-phase and anti-phase bursting in the coupled pre-Bötzinger complex cells

“…Best et al studied a two-cell model network with synaptic coupling and they found that the square-wave bursting is replaced by the top hat bursting (also known as fold/fold cycle bursting) over a wide parameter range (Best et al 2005). The transition from bursting to spiking has been studied in a variety of neuronal models (Bi et al 2014;Duan et al 2012;Rinzel 1985;Izhikevich 2000;Jia et al 2012;Gu and Xiao 2014). The fast-slow decomposition (Rinzel 1985) is an analytic method in the study of bursting activities.…”

Wave emitting and propagation induced by autapse in a forward feedback neuronal network

“…The electric activities of neurons can be observed in biological experiments [6] but also can be reproduced from theoretical neuron models [7] or neuronal circuit [8]. For example, quiescent, spiking, bursting states can be numerically observed in the neuron model under appropriate bifurcation parameters [9,10]. A single and isolate neuron can emerge diverse modes in electric activity by imposing appropriate external forcing current or selecting appropriate intrinsic bifurcation parameter, and transition of electric activity occurs by carefully changing the external forcing current monotonously.…”