Pairing cellular and synaptic dynamics into building blocks of rhythmic neural circuits. A tutorial.

In this study we focus on two subnetworks common in the circuitry of swim central pattern generators (CPGs) in the sea slugs, Melibe leonina and Dendronotus iris and show that they are independently capable of stably producing emergent network bursting. This observation raises the question of whether the coordination of redundant bursting mechanisms plays a role in the generation of rhythm and its regulation in the given swim CPGs. To address this question, we investigate two pairwise rhythm-generating networks and examine the properties of their fundamental components: cellular and synaptic, which are crucial for proper network assembly and its stable function. We perform a slow-fast decomposition analysis of cellular dynamics and highlight its significant bifurcations occurring in isolated and coupled neurons. A novel model for slow synapses with high filtering efficiency and temporal delay is also introduced and examined. Our findings demonstrate the existence of two modes of oscillation in bicellular rhythm-generating networks with network hysteresis: i) a half-center oscillator and ii) an excitatory-inhibitory pair. These 2-cell networks offer potential as common building blocks combined in modular organization of larger neural circuits preserving robust network hysteresis.

Error Function Optimization to Compare Neural Activity and Train Blended Rhythmic Networks.

We present a novel set of quantitative measures for "likeness" (error function) designed to alleviate the time-consuming and subjective nature of manually comparing biological recordings from electrophysiological experiments with the outcomes of their mathematical models. Our innovative "blended" system approach offers an objective, high-throughput, and computationally efficient method for comparing biological and mathematical models. This approach involves using voltage recordings of biological neurons to drive and train mathematical models, facilitating the derivation of the error function for further parameter optimization. Our calibration process incorporates measurements such as action potential (AP) frequency, voltage moving average, voltage envelopes, and the probability of post-synaptic channels. To assess the effectiveness of our method, we utilized the sea slug Melibe leonina swim central pattern generator (CPG) as our model circuit and conducted electrophysiological experiments with TTX to isolate CPG interneurons. During the comparison of biological recordings and mathematically simulated neurons, we performed a grid search of inhibitory and excitatory synapse conductance. Our findings indicate that a weighted sum of simple functions is essential for comprehensively capturing a neuron's rhythmic activity. Overall, our study suggests that our blended system approach holds promise for enabling objective and high-throughput comparisons between biological and mathematical models, offering significant potential for advancing research in neural circuitry and related fields.

Noise-activated barrier crossing in multiattractor dissipative neural networks.

Noise-activated transitions between coexisting attractors are investigated in a chaotic spiking network. At low noise level, attractor hopping consists of discrete bifurcation events that conserve the memory of initial conditions. When the escape probability becomes comparable to the intrabasin hopping probability, the lifetime of attractors is given by a detailed balance where the less coherent attractors act as a sink for the more coherent ones. In this regime, the escape probability follows an activation law allowing us to assign pseudoactivation energies to limit cycle attractors. These pseudoenergies introduce a useful metric for evaluating the resilience of biological rhythms to perturbations.

Measuring chaos in the Lorenz and Rössler models: Fidelity tests for reservoir computing.

This study focuses on the qualitative and quantitative characterization of chaotic systems with the use of a symbolic description. We consider two famous systems, Lorenz and Rössler models with their iconic attractors, and demonstrate that with adequately chosen symbolic partition, three measures of complexity, such as the Shannon source entropy, the Lempel-Ziv complexity, and the Markov transition matrix, work remarkably well for characterizing the degree of chaoticity and precise detecting stability windows in the parameter space. The second message of this study is to showcase the utility of symbolic dynamics with the introduction of a fidelity test for reservoir computing for simulating the properties of the chaos in both models' replicas. The results of these measures are validated by the comparison approach based on one-dimensional return maps and the complexity measures.

Ordered intricacy of Shilnikov saddle-focus homoclinics in symmetric systems.

Using the technique of Poincaré return maps, we disclose an intricate order of subsequent homoclinic bifurcations near the primary figure-8 connection of the Shilnikov saddle-focus in systems with reflection symmetry. We also reveal admissible shapes of the corresponding bifurcation curves in a parameter space. Their scalability ratio and organization are proven to be universal for such homoclinic bifurcations of higher orders. Two applications with similar dynamics due to the Shilnikov saddle-foci are used to illustrate the theory: a smooth adaptation of the Chua circuit and a 3D normal form.

Design Principles for Central Pattern Generators With Preset Rhythms.

This article is concerned with the design of synthetic central pattern generators (CPGs). Biological CPGs are neural circuits that determine a variety of rhythmic activities, including locomotion, in animals. A synthetic CPG is a network of dynamical elements (here called cells) properly coupled by various synapses to emulate rhythms produced by a biological CPG. We focus on CPGs for locomotion of quadrupeds and present our design approach, based on the principles of nonlinear dynamics, bifurcation theory, and parameter optimization. This approach lets us design the synthetic CPG with a set of desired rhythms and switch between them as the parameter representing the control actions from the brain is varied. The developed four-cell CPG can produce four distinct gaits: walk, trot, gallop, and bound, similar to the mouse locomotion. The robustness and adaptability of the network design principles are verified using different cell and synapse models.

Bottom-up approach to torus bifurcation in neuron models.

We study the quasi-periodicity phenomena occurring at the transition between tonic spiking and bursting activities in exemplary biologically plausible Hodgkin-Huxley type models of individual cells and reduced phenomenological models with slow and fast dynamics. Using the geometric slow-fast dissection and the parameter continuation approach, we show that the transition is due to either the torus bifurcation or the period-doubling bifurcation of a stable periodic orbit on the 2D slow-motion manifold near a characteristic fold. Various torus bifurcations including stable and saddle torus-canards, resonant tori, the co-existence of nested tori, and the torus breakdown leading to the onset of complex and bistable dynamics in such systems are examined too.

Bursting dynamics in the normal and failing hearts.

A failing heart differs from healthy hearts by an array of symptomatic characteristics, including impaired Ca2+ transients, upregulation of Na+/Ca2+ exchanger function, reduction of Ca2+ uptake to sarcoplasmic reticulum, reduced K+ currents, and increased propensity to arrhythmias. While significant efforts have been made in both experimental studies and model development to display the causes of heart failure, the full process of deterioration from a healthy to a failing heart yet remains deficiently understood. In this paper, we analyze a highly detailed mathematical model of mouse ventricular myocytes to disclose the key mechanisms underlying the continual transition towards a state of heart failure. We argue that such a transition can be described in mathematical terms as a sequence of bifurcations that the healthy cells undergo while transforming into failing cells. They include normal action potentials and [Ca2+]i transients, action potential and [Ca2+]i alternans, and bursting behaviors. These behaviors where supported by experimental studies of heart failure. The analysis of this model allowed us to identify that the slow component of the fast Na+ current is a key determining factor for the onset of bursting activity in mouse ventricular myocytes.

The Art of Grid Fields: Geometry of Neuronal Time.

The discovery of grid cells in the entorhinal cortex has both elucidated our understanding of spatial representations in the brain, and germinated a large number of theoretical models regarding the mechanisms of these cells' striking spatial firing characteristics. These models cross multiple neurobiological levels that include intrinsic membrane resonance, dendritic integration, after hyperpolarization characteristics and attractor dynamics. Despite the breadth of the models, to our knowledge, parallels can be drawn between grid fields and other temporal dynamics observed in nature, much of which was described by Art Winfree and colleagues long before the initial description of grid fields. Using theoretical and mathematical investigations of oscillators, in a wide array of mediums far from the neurobiology of grid cells, Art Winfree has provided a substantial amount of research with significant and profound similarities. These theories provide specific inferences into the biological mechanisms and extraordinary resemblances across phenomenon. Therefore, this manuscript provides a novel interpretation on the phenomenon of grid fields, from the perspective of coupled oscillators, postulating that grid fields are the spatial representation of phase resetting curves in the brain. In contrast to prior models of gird cells, the current manuscript provides a sketch by which a small network of neurons, each with oscillatory components can operate to form grid cells, perhaps providing a unique hybrid between the competing attractor neural network and oscillatory interference models. The intention of this new interpretation of the data is to encourage novel testable hypotheses.

Robust design of polyrhythmic neural circuits.

Neural circuit motifs producing coexistent rhythmic patterns are treated as building blocks of multifunctional neuronal networks. We study the robustness of such a motif of inhibitory model neurons to reliably sustain bursting polyrhythms under random perturbations. Without noise, the exponential stability of each of the coexisting rhythms increases with strengthened synaptic coupling, thus indicating an increased robustness. Conversely, after adding noise we find that noise-induced rhythm switching intensifies if the coupling strength is increased beyond a critical value, indicating a decreased robustness. We analyze this stochastic arrhythmia and develop a generic description of its dynamic mechanism. Based on our mechanistic insight, we show how physiological parameters of neuronal dynamics and network coupling can be balanced to enhance rhythm robustness against noise. Our findings are applicable to a broad class of relaxation-oscillator networks, including Fitzhugh-Nagumo and other Hodgkin-Huxley-type networks.

Key bifurcations of bursting polyrhythms in 3-cell central pattern generators.

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.

Spontaneous voltage oscillations and response dynamics of a Hodgkin-Huxley type model of sensory hair cells.

We employ a Hodgkin-Huxley type model of basolateral ionic currents in bullfrog saccular hair cells to study the genesis of spontaneous voltage oscillations and their role in shaping the response of the hair cell to external mechanical stimuli. Consistent with recent experimental reports, we find that the spontaneous dynamics of the model can be categorized using conductance parameters of calcium activated potassium, inward rectifier potassium, and mechano-electrical transduction ionic currents. The model is demonstrated to exhibit a broad spectrum of autonomous rhythmic activity, including periodic and quasiperiodic oscillations with two independent frequencies as well as various regular and chaotic bursting patterns. Complex patterns of spontaneous oscillations in the model emerge at small values of the conductance of Ca(2+) activated potassium currents. These patterns are significantly affected by thermal fluctuations of the mechano-electrical transduction current. We show that self-sustained regular voltage oscillations lead to enhanced and sharply tuned sensitivity of the hair cell to weak mechanical periodic stimuli. While regimes of chaotic oscillations are argued to result in poor tuning to sinusoidal driving, chaotically oscillating cells do provide a high sensitivity to low-frequency variations of external stimuli.

Variability of bursting patterns in a neuron model in the presence of noise.

Spiking and bursting patterns of neurons are characterized by a high degree of variability. A single neuron can demonstrate endogenously various bursting patterns, changing in response to external disturbances due to synapses, or to intrinsic factors such as channel noise. We argue that in a model of the leech heart interneuron existing variations of bursting patterns are significantly enhanced by a small noise. In the absence of noise this model shows periodic bursting with fixed numbers of interspikes for most parameter values. As the parameter of activation kinetics of a slow potassium current is shifted to more hyperpolarized values of the membrane potential, the model undergoes a sequence of incremental spike adding transitions accumulating towards a periodic tonic spiking activity. Within a narrow parameter window around every spike adding transition, spike alteration of bursting is deterministically chaotic due to homoclinic bifurcations of a saddle periodic orbit. We have found that near these transitions the interneuron model becomes extremely sensitive to small random perturbations that cause a wide expansion and overlapping of the chaotic windows. The chaotic behavior is characterized by positive values of the largest Lyapunov exponent, and of the Shannon entropy of probability distribution of spike numbers per burst. The windows of chaotic dynamics resemble the Arnold tongues being plotted in the parameter plane, where the noise intensity serves as a second control parameter. We determine the critical noise intensities above which the interneuron model generates only irregular bursting within the overlapped windows.