Rhythms of the nervous system: mathematical themes and variations

The nervous system displays a variety of rhythms in both waking and sleep. These rhythms have been closely associated with different behavioral and cognitive states, but it is still unknown how the nervous system makes use of these rhythms to perform functionally important tasks. To address those questions, it is first useful to understood in a mechanistic way the origin of the rhythms, their interactions, the signals which create the transitions among rhythms, and the ways in which rhythms filter the signals to a network of neurons. This talk discusses how dynamical systems have been used to investigate the origin, properties and interactions of rhythms in the nervous system. It focuses on how the underlying physiology of the cells and synapses of the networks shape the dynamics of the network in different contexts, allowing the variety of dynamical behaviors to be displayed by the same network. The work is presented using a series of related case studies on different rhythms. These case studies are chosen to highlight mathematical issues, and suggest further mathematical work to be done. The topics include: different roles of excitation and inhibition in creating synchronous assemblies of cells, different kinds of building blocks for neural oscillations, and transitions among rhythms. The mathematical issues include reduction of large networks to low dimensional maps, role of noise, global bifurcations, use of probabilistic formulations.Published versio

Controlled generation of switching dynamics among metastable states in pulse-coupled oscillator networks

This research was supported by the Aihara Project, the FIRST program from JSPS, initiated by CSTP, and CREST, JST. Y.C.L. was supported by ARO under Grant No. W911NF-14-1-0504. Z.C.D. was supported by the National Natural Science Foundation of China (No. 11432010). H.L.Z. was supported by “The Fundamental Research Funds for the Central Universities” (No. 3102014JCQ01036), and by the National Natural Science Foundation of China (No. 11502200). We also thank anonymous reviewers for their insightful and useful comments.Peer reviewedPublisher PD

Central pattern generator for swimming in Melibe

The nudibranch mollusc Melibe leonina swims by bending from side to side. We have identified a network of neurons that appears to constitute the central pattern generator (CPG) for this locomotor behavior, one of only a few such networks to be described in cellular detail. The network consists of two pairs of interneurons, termed `swim interneuron 1\u27 (sint1) and `swim interneuron 2\u27 (sint2), arranged around a plane of bilateral symmetry. Interneurons on one side of the brain, which includes the paired cerebral, pleural and pedal ganglia, coordinate bending movements toward the same side and communicate via non-rectifying electrical synapses. Interneurons on opposite sides of the brain coordinate antagonistic movements and communicate over mutually inhibitory synaptic pathways. Several criteria were used to identify members of the swim CPG, the most important being the ability to shift the phase of swimming behavior in a quantitative fashion by briefly altering the firing pattern of an individual neuron. Strong depolarization of any of the interneurons produces an ipsilateral swimming movement during which the several components of the motor act occur in sequence. Strong hyperpolarization causes swimming to stop and leaves the animal contracted to the opposite side for the duration of the hyperpolarization. The four swim interneurons make appropriate synaptic connections with motoneurons, exciting synergists and inhibiting antagonists. Finally, these are the only neurons that were found to have this set of properties in spite of concerted efforts to sample widely in the Melibe CNS. This led us to conclude that these four cells constitute the CPG for swimming. While sint1 and sint2 work together during swimming, they play different roles in the generation of other behaviors. Sint1 is normally silent when the animal is crawling on a surface but it depolarizes and begins to fire in strong bursts once the foot is dislodged and the animal begins to swim. Sint2 also fires in bursts during swimming, but it is not silent in non-swimming animals. Instead activity in sint2 is correlated with turning movements as the animal crawls on a surface. This suggests that the Melibe motor system is organized in a hierarchy and that the alternating movements characteristic of swimming emerge when activity in sint1 and sint2 is bound together

Spike-Train Responses of a Pair of Hodgkin-Huxley Neurons with Time-Delayed Couplings

Model calculations have been performed on the spike-train response of a pair of Hodgkin-Huxley (HH) neurons coupled by recurrent excitatory-excitatory couplings with time delay. The coupled, excitable HH neurons are assumed to receive the two kinds of spike-train inputs: the transient input consisting of $M$ impulses for the finite duration ($M$: integer) and the sequential input with the constant interspike interval (ISI). The distribution of the output ISI $T_{\rm o}$ shows a rich of variety depending on the coupling strength and the time delay. The comparison is made between the dependence of the output ISI for the transient inputs and that for the sequential inputs.Comment: 19 pages, 4 figure

Novel Modes of Synchronization and Extreme Events in Coupled Chemical Oscillators

We experimentally and computationally investigate dynamical behaviors in coupled chemical oscillators. These networks of chemical oscillators are created using catalytic Ru(bpy)32+ loaded cation exchange beads submerged in catalyst-free Belousov-Zhabotinsky (BZ) solutions. Various network structures are created by utilizing the photosensitive nature of the Ru(bpy) 32+ catalyst. The response of the oscillators due to light stimuli can be characterized by constructing a phase response curve (PRC). The PRC quantifies the excitatory and inhibitory responses of BZ oscillators due to applied light perturbations as a function of the oscillators\u27 phase. Different initial concentrations of reactants in the BZ reaction solutions can vary the degree in the excitatory and inhibitory regions of the PRC. We explore synchronization in star networks in both excitatory and inhibitory systems. We describe experiments, simulations, and analytical theory that provides a detailed characterization of novel modes of synchronization in chemical oscillator networks. Synchronization of peripheral oscillators coupled through a hub oscillator is exhibited at coupling strengths leading to novel synchronization of the hub with the peripheral oscillators. The heterogenous peripheral oscillators have different phase velocities that give rise to divergence; however, the perturbation from the hub acts to realign the phases by delaying the faster oscillators more than the slower oscillators. A theoretical analysis provides insights into the mechanism of the synchronization. Computational studies into extreme events are investigated using a modified four-variable Oregonator model, which describes the BZ system. Extreme events are ubiquitous throughout biological, natural, social, and financial systems. Examples of such events are epileptic seizures, earthquakes, riots, and stock market crashes. These events are considered rare excursions from the normal dynamics of a system, which are considered aperiodic in occurrence. The consequences that these events have on the system makes the development of models and experimental methods to study these events important. We will describe the appearance of extreme events in the Oregonator system using instantaneous and time-delayed coupling. We will also discuss a proposed mechanism for the sudden appearance of extreme events in both instantaneous and time-delayed coupling

Synchronization of Coupled and Periodically Forced Chemical Oscillators

Physiological rhythms are essential in all living organisms. Such rhythms are regulated through the interactions of many cells. Deviation of a biological system from its normal rhythms can lead to physiological maladies. The tremor and symptoms associated with Parkinson\u27s disease are thought to emerge from abnormal synchrony of neuronal activity within the neural network of the brain. Deep brain stimulation is a therapeutic technique that can remove this pathological synchronization by the application of a periodic desynchronizing signal. Herein, we used the photosensitive Belousov--Zhabotinsky (BZ) chemical reaction to test the mechanism of deep brain stimulation. A collection of oscillators are initially synchronized using a regular light signal. Desynchronization is then attempted using an appropriately chosen desynchronizing signal based on information found in the phase response curve.;Coupled oscillators in various network topologies form the most common prototypical systems for studying networks of dynamical elements. In the present study, we couple discrete BZ photochemical oscillators in a network configuration. Different behaviors are observed on varying the coupling strength and the frequency heterogeneity, including incoherent oscillations to partial and full frequency entrainment. Phase clusters are organized symmetrically or non-symmetrically in phase-lag synchronization structures, a novel phase wave entrainment behavior in non-continuous media. The behavior is observed over a range of moderate coupling strengths and a broad frequency distribution of the oscillators