Multimodal behavior in a four neuron ring circuit: mode switching

Controllable switching between stable modes in a small network of pulse-coupled chemical oscillators

Switching between stable oscillatory modes in a network of four Belousov–Zhabotinsky oscillators unidirectionally coupled in a ring analysed computationally and experimentally.

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.

Dynamical regimes of four oscillators with excitatory pulse coupling

Dynamics of four almost identical chemical oscillators pulse coupled via excitatory coupling with time delays are systematically studied.

Dynamical regimes of four almost identical chemical oscillators coupled via pulse inhibitory coupling with time delay

The dynamics of four almost identical pulse coupled chemical oscillators with time delay are systematically studied.

A survey on CPG-inspired control models and system implementation

This paper surveys the developments of the last 20 years in the field of central pattern generator (CPG) inspired locomotion control, with particular emphasis on the fast emerging robotics-related applications. Functioning as a biological neural network, CPGs can be considered as a group of coupled neurons that generate rhythmic signals without sensory feedback; however, sensory feedback is needed to shape the CPG signals. The basic idea in engineering endeavors is to replicate this intrinsic, computationally efficient, distributed control mechanism for multiple articulated joints, or multi-DOF control cases. In terms of various abstraction levels, existing CPG control models and their extensions are reviewed with a focus on the relative advantages and disadvantages of the models, including ease of design and implementation. The main issues arising from design, optimization, and implementation of the CPG-based control as well as possible alternatives are further discussed, with an attempt to shed more light on locomotion control-oriented theories and applications. The design challenges and trends associated with the further advancement of this area are also summarized.

Cellularly-driven differences in network synchronization propensity are differentially modulated by firing frequency

Spatiotemporal pattern formation in neuronal networks depends on the interplay between cellular and network synchronization properties. The neuronal phase response curve (PRC) is an experimentally obtainable measure that characterizes the cellular response to small perturbations, and can serve as an indicator of cellular propensity for synchronization. Two broad classes of PRCs have been identified for neurons: Type I, in which small excitatory perturbations induce only advances in firing, and Type II, in which small excitatory perturbations can induce both advances and delays in firing. Interestingly, neuronal PRCs are usually attenuated with increased spiking frequency, and Type II PRCs typically exhibit a greater attenuation of the phase delay region than of the phase advance region. We found that this phenomenon arises from an interplay between the time constants of active ionic currents and the interspike interval. As a result, excitatory networks consisting of neurons with Type I PRCs responded very differently to frequency modulation compared to excitatory networks composed of neurons with Type II PRCs. Specifically, increased frequency induced a sharp decrease in synchrony of networks of Type II neurons, while frequency increases only minimally affected synchrony in networks of Type I neurons. These results are demonstrated in networks in which both types of neurons were modeled generically with the Morris-Lecar model, as well as in networks consisting of Hodgkin-Huxley-based model cortical pyramidal cells in which simulated effects of acetylcholine changed PRC type. These results are robust to different network structures, synaptic strengths and modes of driving neuronal activity, and they indicate that Type I and Type II excitatory networks may display two distinct modes of processing information.

Synchronization of the firing of neurons in the brain is related to many cognitive functions, such as recognizing faces, discriminating odors, and coordinating movement. It is therefore important to understand what properties of neuronal networks promote synchrony of neural firing. One measure that is often used to determine the contribution of individual neurons to network synchrony is called the phase response curve (PRC). PRCs describe how the timing of neuronal firing changes depending on when input, such as a synaptic signal, is received by the neuron. A characteristic of PRCs that has previously not been well understood is that they change dramatically as the neuron's firing frequency is modulated. This effect carries potential significance, since cognitive functions are often associated with specific frequencies of network activity in the brain. We showed computationally that the frequency dependence of PRCs can be explained by the relative timing of ionic membrane currents with respect to the time between spike firings. Our simulations also showed that the frequency dependence of neuronal PRCs leads to frequency-dependent changes in network synchronization that can be different for different neuron types. These results further our understanding of how synchronization is generated in the brain to support various cognitive functions.

Pulse coupled oscillators and the phase resetting curve

Limit cycle oscillators that are coupled in a pulsatile manner are referred to as pulse coupled oscillators. In these oscillators, the interactions take the form of brief pulses such that the effect of one input dies out before the next is received. A phase resetting curve (PRC) keeps track of how much an input advances or delays the next spike in an oscillatory neuron depending upon where in the cycle the input is applied. PRCs can be used to predict phase locking in networks of pulse coupled oscillators. In some studies of pulse coupled oscillators, a specific form is assumed for the interactions between oscillators, but a more general approach is to formulate the problem assuming a PRC that is generated using a perturbation that approximates the input received in the real biological network. In general, this approach requires that circuit architecture and a specific firing pattern be assumed. This allows the construction of discrete maps from one event to the next. The fixed points of these maps correspond to periodic firing modes and are easier to locate and analyze for stability compared to locating and analyzing periodic modes in the original network directly. Alternatively, maps based on the PRC have been constructed that do not presuppose a firing order. Specific circuits that have been analyzed under the assumption of pulsatile coupling include one to one lockings in a periodically forced oscillator or an oscillator forced at a fixed delay after a threshold event, two bidirectionally coupled oscillators with and without delays, a unidirectional N-ring of oscillators, and N all-to-all networks.

Phase resetting curves allow for simple and accurate prediction of robust N:1 phase locking for strongly coupled neural oscillators

Existence and stability criteria for harmonic locking modes were derived for two reciprocally pulse coupled oscillators based on their first and second order phase resetting curves. Our theoretical methods are general in the sense that no assumptions about the strength of coupling, type of synaptic coupling, and model are made. These methods were then tested using two reciprocally inhibitory Wang and Buzsáki model neurons. The existence of bands of 2:1, 3:1, 4:1, and 5:1 phase locking in the relative frequency parameter space was predicted correctly, as was the phase of the slow neuron's spike within the cycle of the fast neuron in which it occurred. For weak coupling the bands are very narrow, but strong coupling broadens the bands. The predictions of the pulse coupled method agreed with weak coupling methods in the weak coupling regime, but extended predictability into the strong coupling regime. We show that our prediction method generalizes to pairs of neural oscillators coupled through excitatory synapses, and to networks of multiple oscillatory neurons. The main limitation of the method is the central assumption that the effect of each input dies out before the next input is received.

Phase-resetting curves determine synchronization, phase locking, and clustering in networks of neural oscillators

Networks of model neurons were constructed and their activity was predicted using an iterated map based solely on the phase-resetting curves (PRCs). The predictions were quite accurate provided that the resetting to simultaneous inputs was calculated using the sum of the simultaneously active conductances, obviating the need for weak coupling assumptions. Fully synchronous activity was observed only when the slope of the PRC at a phase of zero, corresponding to spike initiation, was positive. A novel stability criterion was developed and tested for all-to-all networks of identical, identically connected neurons. When the PRC generated using N-1 simultaneously active inputs becomes too steep, the fully synchronous mode loses stability in a network of N model neurons. Therefore, the stability of synchrony can be lost by increasing the slope of this PRC either by increasing the network size or the strength of the individual synapses. Existence and stability criteria were also developed and tested for the splay mode in which neurons fire sequentially. Finally, N/M synchronous subclusters of M neurons were predicted using the intersection of parameters that supported both between-cluster splay and within-cluster synchrony. Surprisingly, the splay mode between clusters could enforce synchrony on subclusters that were incapable of synchronizing themselves. These results can be used to gain insights into the activity of networks of biological neurons whose PRCs can be measured.

Predictions of phase-locking in excitatory hybrid networks: excitation does not promote phase-locking in pattern-generating networks as reliably as inhibition

Phase-locked activity is thought to underlie many high-level functions of the nervous system, the simplest of which are produced by central pattern generators (CPGs). It is not known whether we can define a theoretical framework that is sufficiently general to predict phase-locking in actual biological CPGs, nor is it known why the CPGs that have been characterized are dominated by inhibition. Previously, we applied a method based on phase response curves measured using inputs of biologically realistic amplitude and duration to predict the existence and stability of 1:1 phase-locked modes in hybrid networks of one biological and one model bursting neuron reciprocally connected with artificial inhibitory synapses. Here we extend this analysis to excitatory coupling. Using the pyloric dilator neuron from the stomatogastric ganglion of the American lobster as our biological cell, we experimentally prepared 86 networks using five biological neurons, four model neurons, and heterogeneous synapse strengths between 1 and 10,000 nS. In 77% of networks, our method was robust to biological noise and accurately predicted the phasic relationships. In 3%, our method was inaccurate. The remaining 20% were not amenable to analysis because our theoretical assumptions were violated. The high failure rate for excitation compared with inhibition was due to differential effects of noise and feedback on excitatory versus inhibitory coupling and suggests that CPGs dominated by excitatory synapses would require precise tuning to function, which may explain why CPGs rely primarily on inhibitory synapses.

Multifunctional pattern-generating circuits

The ability of distinct anatomical circuits to generate multiple behavioral patterns is widespread among vertebrate and invertebrate species. These multifunctional neuronal circuits are the result of multistable neural dynamics and modular organization. The evidence suggests multifunctional circuits can be classified by distinct architectures, yet the activity patterns of individual neurons involved in more than one behavior can vary dramatically. Several mechanisms, including sensory input, the parallel activity of projection neurons, neuromodulation, and biomechanics, are responsible for the switching between patterns. Recent advances in both analytical and experimental tools have aided the study of these complex circuits.

Using phase resetting to predict 1:1 and 2:2 locking in two neuron networks in which firing order is not always preserved

Our goal is to understand how nearly synchronous modes arise in heterogenous networks of neurons. In heterogenous networks, instead of exact synchrony, nearly synchronous modes arise, which include both 1:1 and 2:2 phase-locked modes. Existence and stability criteria for 2:2 phase-locked modes in reciprocally coupled two neuron circuits were derived based on the open loop phase resetting curve (PRC) without the assumption of weak coupling. The PRC for each component neuron was generated using the change in synaptic conductance produced by a presynaptic action potential as the perturbation. Separate derivations were required for modes in which the firing order is preserved and for those in which it alternates. Networks composed of two model neurons coupled by reciprocal inhibition were examined to test the predictions. The parameter regimes in which both types of nearly synchronous modes are exhibited were accurately predicted both qualitatively and quantitatively provided that the synaptic time constant is short with respect to the period and that the effect of second order resetting is considered. In contrast, PRC methods based on weak coupling could not predict 2:2 modes and did not predict the 1:1 modes with the level of accuracy achieved by the strong coupling methods. The strong coupling prediction methods provide insight into what manipulations promote near-synchrony in a two neuron network and may also have predictive value for larger networks, which can also manifest changes in firing order. We also identify a novel route by which synchrony is lost in mildly heterogenous networks.

Design and implementation of multipattern generators in analog VLSI

In recent years, computational biologists have shown through simulation that small neural networks with fixed connectivity are capable of producing multiple output rhythms in response to transient inputs. It is believed that such networks may play a key role in certain biological behaviors such as dynamic gait control. In this paper, we present a novel method for designing continuous-time recurrent neural networks (CTRNNs) that contain multiple embedded limit cycles, and we show that it is possible to switch the networks between these embedded limit cycles with simple transient inputs. We also describe the design and testing of a fully integrated four-neuron CTRNN chip that is used to implement the neural network pattern generators. We provide two example multipattern generators and show that the measured waveforms from the chip agree well with numerical simulations.

Phase resetting and phase locking in hybrid circuits of one model and one biological neuron

To determine why elements of central pattern generators phase lock in a particular pattern under some conditions but not others, we tested a theoretical pattern prediction method. The method is based on the tabulated open loop pulsatile interactions of bursting neurons on a cycle-by-cycle basis and was tested in closed loop hybrid circuits composed of one bursting biological neuron and one bursting model neuron coupled using the dynamic clamp. A total of 164 hybrid networks were formed by varying the synaptic conductances. The prediction of 1:1 phase locking agreed qualitatively with the experimental observations, except in three hybrid circuits in which 1:1 locking was predicted but not observed. Correct predictions sometimes required consideration of the second order phase resetting, which measures the change in the timing of the second burst after the perturbation. The method was robust to offsets between the initiation of bursting in the presynaptic neuron and the activation of the synaptic coupling with the postsynaptic neuron. The quantitative accuracy of the predictions fell within the variability (10%) in the experimentally observed intrinsic period and phase resetting curve (PRC), despite changes in the burst duration of the neurons between open and closed loop conditions.