Key Bifurcations of Bursting Polyrhythms in 3-Cell Central Pattern Generators

Abstract

We identify and describe the key qualitative rhythmic states in various 3-cell network motifs of a multifunctional central pattern generator (CPG). Such CPGs are neural microcircuits of cells whose synergetic interactions produce multiple states with distinct phase-locked patterns of bursting activity. To study biologically plausible CPG models, we develop a suite of computational tools that reduce the problem of stability and existence of rhythmic patterns in networks to the bifurcation analysis of fixed points and invariant curves of a Poincaré return maps for phase lags between cells. We explore different functional possibilities for motifs involving symmetry breaking and heterogeneity. This is achieved by varying coupling properties of the synapses between the cells and studying the qualitative changes in the structure of the corresponding return maps. Our findings provide a systematic basis for understanding plausible biophysical mechanisms for the regulation of rhythmic patterns generated by various CPGs in the context of motor control such as gait-switching in locomotion. Our analysis does not require knowledge of the equations modeling the system and provides a powerful qualitative approach to studying detailed models of rhythmic behavior. Thus, our approach is applicable to a wide range of biological phenomena beyond motor control.

Robust design of polyrhythmic neural circuits

“…We find that strengthening synaptic coupling fulfills a dual role: at weak coupling the stability increases against gradual dephasing of bursting patterns [43,46,50], but at strong coupling the rhythm robustness decreases due to sensitization of the vulnerable phase. This duality is due to a coupling-dependent soft-to hard-lock transition.…”

Linking dynamics of the inhibitory network to the input structure

“…It is worth noting that similar networks of inhibitory neurons have been extensively studied before, especially in the context of locomotion and CPG dynamics (Golubitsky et al 1998; Collins and bifurcation 1992; Nowotny and Rabinovich 2007; Belykh and Shilnikov 2008; Schwabedal et al 2014; Wojcik et al 2014). In general, networks of identical neurons display coexisting oscillatory rhythms, which are characterized by distinct phase-locking patterns between individual neurons.…”

“…2b) commonly exhibit a prominent multistability (e.g. see (Wojcik et al 2014)), which manifests itself as a coexistence of a certain number of possible stable solutions (attractors). Indeed, detailed description of similar case (relatively small |Δ I |) in Section 3.3 revealed bistability of two possible bursting waves (clockwise and counter- clockwise), which implies that the proper initial conditions are required to initiate the “correct” bursting sequence.…”

“…In order to describe all possible stable solutions, we used Poincare map. Our methods are adaptation of the technique used in (Wojcik et al 2014) for theoretical description of the central pattern generator dynamics. As a Poincare section we select the manifold P = l 12 ∪ l 13 (illustration is in the Fig.…”

Coupling functions: Universal insights into dynamical interaction mechanisms

“…1) and to good effect: in chemistry, for understanding, effecting, or predicting interactions between oscillatory electrochemical reactions (Miyazaki and Kinoshita, 2006;Kiss et al, 2007;Tokuda et al, 2007;Blaha et al, 2011;Kori et al, 2014); in cardiorespiratory physiology Iatsenko et al, 2013;Ticcinelli et al, 2017) for reconstruction of the human cardiorespiratory coupling function and phase resetting curve, for assessing cardiorespiratory time variability, and for studying the evolution of the cardiorespiratory coupling functions with age; in neuroscience for revealing the cross-frequency coupling functions between neural oscillations (Stankovski et al, 2015); and polyrithmic behavior in neuronal circuits (Wojcik et al, 2014;Schwabedal, Knapper, and Shilnikov, 2016); in social sciences for determining the function underlying the interactions between democracy and economic growth ; for mechanical interactions between coupled metronomes (Kralemann et al, 2008); and in secure communications where a new protocol was developed explicitly based on amplitude coupling functions (Stankovski, McClintock, and Stefanovska, 2014).…”