Bistability of bursting and silence regimes in a model of a leech heart interneuron

Control of transitions between locomotor-like and paw shake-like rhythms in a model of a multistable central pattern generator

The ability of the same neuronal circuit to control different motor functions is an actively debated concept. Previously, we showed in a model that a single multistable central pattern generator (CPG) could produce two different rhythmic motor patterns, slow and fast, corresponding to cat locomotion and paw shaking. A locomotor-like rhythm (~1 Hz) and a paw shake-like rhythm (~10 Hz) did coexist in our model, and, by applying a single pulse of current, we could switch the CPG from one regime to another (Bondy B, Klishko AN, Edwards DH, Prilutsky BI, Cymbalyuk G. In: Neuromechanical Modeling of Posture and Locomotion, 2016). Here we investigated the roles of slow intrinsic ionic currents in this multistability. The CPG is modeled as a half-center oscillator circuit comprising two reciprocally inhibitory neurons. Each neuron is equipped with two slow inward currents, a Na⁺ current ( I_NaS) and a Ca²⁺ current ( I_CaS). I_CaS inactivates much more slowly and at more hyperpolarized voltages than I_NaS. We demonstrate that I_NaS is the primary current driving the paw shake-like bursting. I_CaS is crucial for the locomotor-like bursting, and it is inactivated during the paw shake-like activity. We investigate the sensitivity of the bursting regimes to perturbations, using a pulse of current to induce a switch from one regime to the other, and we demonstrate that the transition duration is dependent on pulse amplitude and application phase. We also investigate the modulatory roles of the strength of various currents on characteristics of these rhythms and show that their effects are regime specific. We conclude that a multistable CPG is physiologically plausible and derive testable predictions of the model. NEW & NOTEWORTHY Little is known about how a single central pattern generator could produce multiple rhythms. We describe a novel mechanism for multistability of bursting regimes with strongly distinct periods. The proposed mechanism emphasizes the role of intrinsic cellular dynamics over synaptic dynamics in the production of multistability. We describe how the temporal characteristics of multiple rhythms could be controlled by neuromodulation and how single pulses of current could produce a switch between regimes in a functional fashion.

Propensity for Bistability of Bursting and Silence in the Leech Heart Interneuron

The coexistence of neuronal activity regimes has been reported under normal and pathological conditions. Such multistability could enhance the flexibility of the nervous system and has many implications for motor control, memory, and decision making. Multistability is commonly promoted by neuromodulation targeting specific membrane ionic currents. Here, we investigated how modulation of different ionic currents could affect the neuronal propensity for bistability. We considered a leech heart interneuron model. It exhibits bistability of bursting and silence in a narrow range of the leak current parameters, conductance (g_leak) and reversal potential (E_leak). We assessed the propensity for bistability of the model by using bifurcation diagrams. On the diagram (g_leak, E_leak), we mapped bursting and silent regimes. For the canonical value of E_leak we determined the range of g_leak which supported the bistability. We use this range as an index of propensity for bistability. We investigated how this index was affected by alterations of ionic currents. We systematically changed their conductances, one at a time, and built corresponding bifurcation diagrams in parameter planes of the maximal conductance of a given current and the leak conductance. We found that conductance of only one current substantially affected the index of propensity; the increase of the maximal conductance of the hyperpolarization-activated cationic current increased the propensity index. The second conductance with the strongest effect was the conductance of the low-threshold fast Ca²⁺ current; its reduction increased the propensity index although the effect was about two times smaller in magnitude. Analyzing the model with both changes applied simultaneously, we found that the diagram (g_leak, E_leak) showed a progressively expanded area of bistability of bursting and silence.

A codimension-2 bifurcation controlling endogenous bursting activity and pulse-triggered responses of a neuron model

The dynamics of individual neurons are crucial for producing functional activity in neuronal networks. An open question is how temporal characteristics can be controlled in bursting activity and in transient neuronal responses to synaptic input. Bifurcation theory provides a framework to discover generic mechanisms addressing this question. We present a family of mechanisms organized around a global codimension-2 bifurcation. The cornerstone bifurcation is located at the intersection of the border between bursting and spiking and the border between bursting and silence. These borders correspond to the blue sky catastrophe bifurcation and the saddle-node bifurcation on an invariant circle (SNIC) curves, respectively. The cornerstone bifurcation satisfies the conditions for both the blue sky catastrophe and SNIC. The burst duration and interburst interval increase as the inverse of the square root of the difference between the corresponding bifurcation parameter and its bifurcation value. For a given set of burst duration and interburst interval, one can find the parameter values supporting these temporal characteristics. The cornerstone bifurcation also determines the responses of silent and spiking neurons. In a silent neuron with parameters close to the SNIC, a pulse of current triggers a single burst. In a spiking neuron with parameters close to the blue sky catastrophe, a pulse of current temporarily silences the neuron. These responses are stereotypical: the durations of the transient intervals-the duration of the burst and the duration of latency to spiking-are governed by the inverse-square-root laws. The mechanisms described here could be used to coordinate neuromuscular control in central pattern generators. As proof of principle, we construct small networks that control metachronal-wave motor pattern exhibited in locomotion. This pattern is determined by the phase relations of bursting neurons in a simple central pattern generator modeled by a chain of oscillators.

Improving noise resistance of intrinsic rhythms in a square-wave burster model

The square-wave burster (Wang and Rinzel, 2003) is a class of autonomous bursting cells that share a bifurcation structure. It is known that this class of cells is involved in the generation of various life-supporting rhythms. In our research to realize an electronic circuit that mimics the rhythm generating mechanism in the square-wave burster, our circuit experimentally exhibited severe fluctuations in its rhythmic activity. We have found a noise-sensitive region in the phase portrait of the ideal model and have proposed modifications of the model that can reduce this fluctuation. A possible modification to ionic-conductance neuron models (Kohno and Aihara, 2011) was inspired by them. This modification, however, cannot be applied to a group of square-wave bursters, including the Butera-Rinzel-Smith model (Butera et al., 1999; Del Negro et al., 2001), which is a model of the pre-Bötzinger complex bursting neuron that plays a crucial role in the generation of respiration rhythms, because this modification premises that the slow dynamics originates from an activation gate variable of a hyperpolarizing ionic current. However, in some square-wave bursters, they are controlled by an inactivation gate variable of a depolarizing ionic current. In this study, we proposed a similar modification with a completely different mechanism that can be applied to this group of square-wave bursters. In the presence of noises, the modified Butera-Rinzel-Smith model can generate rhythmic activity that is more stable and similar to biological observations than the original model. The mechanisms underlying this modification are explained with noisy bifurcation diagrams.

High prevalence of multistability of rest states and bursting in a database of a model neuron

Flexibility in neuronal circuits has its roots in the dynamical richness of their neurons. Depending on their membrane properties single neurons can produce a plethora of activity regimes including silence, spiking and bursting. What is less appreciated is that these regimes can coexist with each other so that a transient stimulus can cause persistent change in the activity of a given neuron. Such multistability of the neuronal dynamics has been shown in a variety of neurons under different modulatory conditions. It can play either a functional role or present a substrate for dynamical diseases. We considered a database of an isolated leech heart interneuron model that can display silent, tonic spiking and bursting regimes. We analyzed only the cases of endogenous bursters producing functional half-center oscillators (HCOs). Using a one parameter (the leak conductance ()) bifurcation analysis, we extended the database to include silent regimes (stationary states) and systematically classified cases for the coexistence of silent and bursting regimes. We showed that different cases could exhibit two stable depolarized stationary states and two hyperpolarized stationary states in addition to various spiking and bursting regimes. We analyzed all cases of endogenous bursters and found that 18% of the cases were multistable, exhibiting coexistences of stationary states and bursting. Moreover, 91% of the cases exhibited multistability in some range of . We also explored HCOs built of multistable neuron cases with coexisting stationary states and a bursting regime. In 96% of cases analyzed, the HCOs resumed normal alternating bursting after one of the neurons was reset to a stationary state, proving themselves robust against this perturbation.

It is often not appreciated that different activity regimes can coexist with each other in a given neuron so that a transient stimulus can cause a persistent change of activity. Such multistability of the neuronal dynamics has in fact been shown in a variety of neurons and can play either a functional role or present a substrate for neurological diseases. We explored the propensity for multistability in a database of a leech heart interneuron model, testing each case (parameter set) in a database for multistability. We found a large proportion of multistable cases, especially the coexistence of silent and bursting regimes. This was a surprising result, since these cells pace the heartbeat of the leech, and the coexistence of silence and bursting could disrupt the functional pattern, threatening the viability of the leech. Analysis of networks of mutually inhibitory multistable neurons, however, showed robustness in maintaining functional activity, suggesting that the mutually inhibitory coupling can act as a protective mechanism against failures induced by multistability.