<p>Pattern Formation in Coupled Networks with Inhibition and Gap Junctions</p>

In this dissertation we analyze networks of coupled phase oscillators. We consider systems where long range chemical coupling and short range electrical coupling have opposite effects on the synchronization process. We look at the existence and stability of three patterns of activity: synchrony, clustered state and asynchrony. In Chapter 1, we develop a minimal phase model using experimental results for the olfactory system of Limax. We study the synchronous solution as the strength of synaptic coupling increases. We explain the emergence of traveling waves in the system without a frequency gradient. We construct the normal form for the pitchfork bifurcation and compare our analytical results with numerical simulations. In Chapter 2, we study a mean-field coupled network of phase oscillators for which a stable two-cluster solution exists. The addition of nearest neighbor gap junction coupling destroys the stability of the cluster solution. When the gap junction coupling is strong there is a series of traveling wave solutions depending on the size of the network. We see bistability in the system between clustered state, periodic solutions and traveling waves. The bistability properties also change with the network size. We analyze the system numerically and analytically. In Chapter 3, we turn our attention to a very popular model about network synchronization. We represent the Kuramoto model in its original form and calculate the main results using a different technique. We also look at a modified version and study how this effects synchronization. We consider a collection of oscillators organized in m groups. The addition of gap junctions creates a wave like behavior

Simulation and robotics studies of salamander locomotion: Applying neurobiological principles to the control of locomotion in robots

This article presents a project that aims at understanding the neural circuitry controlling salamander locomotion, and developing an amphibious salamander-like robot capable of replicating its bimodal locomotion, namely swimming and terrestrial walking. The controllers of the robot are central pattern generator models inspired by the salamander's locomotion control network. The goal of the project is twofold: (1) to use robots as tools for gaining a better understanding of locomotion control in vertebrates and (2) to develop new robot and control technologies for developing agile and adaptive outdoor robots. The article has four parts. We first describe the motivations behind the project. We then present neuromechanical simulation studies of locomotion control in salamanders. This is followed by a description of the current stage of the robotic developments. We conclude the article with a discussion on the usefulness of robots in neuroscience research with a special focus on locomotion contro

Loss of synchrony in an inhibitory network of type-I oscillators

Synchronization of excitable cells coupled by reciprocal inhibition is a topic of significant interest due to the important role that inhibitory synaptic interaction plays in the generation and regulation of coherent rhythmic activity in a variety of neural systems. While recent work revealed the synchronizing influence of inhibitory coupling on the dynamics of many networks, it is known that strong coupling can destabilize phase-locked firing. Here we examine the loss of synchrony caused by an increase in inhibitory coupling in networks of type-I Morris-Lecar model oscillators, which is characterized by a period-doubling cascade and leads to mode-locked states with alternation in the firing order of the two cells, as reported recently by Maran and Canavier (2007) for a network of Wang-Buzsáki model neurons. Although alternating- order firing has been previously reported as a near-synchronous state, we show that the stable phase difference between the spikes of the two Morris-Lecar cells can constitute as much as 70% of the unperturbed oscillation period. Further, we examine the generality of this phenomenon for a class of type-I oscillators that are close to their excitation thresholds, and provide an intuitive geometric description of such leap-frog dynamics. In the Morris-Lecar model network, the alternation in the firing order arises under the condition of fast closing of K+ channels at hyperpolarized potentials, which leads to slow dynamics of membrane potential upon synaptic inhibition, allowing the presynaptic cell to advance past the postsynaptic cell in each cycle of the oscillation. Further, we show that non-zero synaptic decay time is crucial for the existence of leap-frog firing in networks of phase oscillators. However, we demonstrate that leap-frog spiking can also be obtained in pulse-coupled inhibitory networks of one-dimensional oscillators with a multi-branched phase domain, for instance in a network of quadratic integrate-and-fire model cells. Also, we show that the entire bifurcation structure of the network can be explained by a simple scaling of the STRC (spike- time response curve) amplitude, using a simplified quadratic STRC as an example, and derive the general conditions on the shape of the STRC function that leads to leap-frog firing. Further, for the case of a homogeneous network, we establish quantitative conditions on the phase resetting properties of each cell necessary for stable alternating-order spiking, complementing the analysis of Goel and Ermentrout (2002) of the order-preserving phase transition map. We show that the extension of STRC to negative values of phase is necessary to predict the response of a model cell to several close non-weak perturbations. This allows us for instance to accurately describe the dynamics of non-weakly coupled network of three model cells. Finally, the phase return map is also extended to the heterogenous network, and is used to analyze both the order-alternating firing and the order-preserving non-zero phase locked state in this case

Mathematical frameworks for oscillatory network dynamics in neuroscience

The tools of weakly coupled phase oscillator theory have had a profound impact on the neuroscience community, providing insight into a variety of network behaviours ranging from central pattern generation to synchronisation, as well as predicting novel network states such as chimeras. However, there are many instances where this theory is expected to break down, say in the presence of strong coupling, or must be carefully interpreted, as in the presence of stochastic forcing. There are also surprises in the dynamical complexity of the attractors that can robustly appear—for example, heteroclinic network attractors. In this review we present a set of mathemat- ical tools that are suitable for addressing the dynamics of oscillatory neural networks, broadening from a standard phase oscillator perspective to provide a practical frame- work for further successful applications of mathematics to understanding network dynamics in neuroscience

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

Effect of small-world topology on wave propagation on networks of excitable elements

We study excitation waves on a Newman-Watts small-world network model of coupled excitable elements. Depending on the global coupling strength, we find differing resilience to the added long-range links and different mechanisms of propagation failure. For high coupling strengths, we show agreement between the network and a reaction-diffusion model with additional mean-field term. Employing this approximation, we are able to estimate the critical density of long-range links for propagation failure.Comment: 19 pages, 8 figures and 5 pages supplementary materia

A dendritic mechanism for decoding traveling waves: Principles and applications to motor cortex

Traveling waves of neuronal oscillations have been observed in many cortical regions, including the motor and sensory cortex. Such waves are often modulated in a task-dependent fashion although their precise functional role remains a matter of debate. Here we conjecture that the cortex can utilize the direction and wavelength of traveling waves to encode information. We present a novel neural mechanism by which such information may be decoded by the spatial arrangement of receptors within the dendritic receptor field. In particular, we show how the density distributions of excitatory and inhibitory receptors can combine to act as a spatial filter of wave patterns. The proposed dendritic mechanism ensures that the neuron selectively responds to specific wave patterns, thus constituting a neural basis of pattern decoding. We validate this proposal in the descending motor system, where we model the large receptor fields of the pyramidal tract neurons — the principle outputs of the motor cortex — decoding motor commands encoded in the direction of traveling wave patterns in motor cortex. We use an existing model of field oscillations in motor cortex to investigate how the topology of the pyramidal cell receptor field acts to tune the cells responses to specific oscillatory wave patterns, even when those patterns are highly degraded. The model replicates key findings of the descending motor system during simple motor tasks, including variable interspike intervals and weak corticospinal coherence. By additionally showing how the nature of the wave patterns can be controlled by modulating the topology of local intra-cortical connections, we hence propose a novel integrated neuronal model of encoding and decoding motor commands

Dynamical principles in neuroscience

Dynamical modeling of neural systems and brain functions has a history of success over the last half century. This includes, for example, the explanation and prediction of some features of neural rhythmic behaviors. Many interesting dynamical models of learning and memory based on physiological experiments have been suggested over the last two decades. Dynamical models even of consciousness now exist. Usually these models and results are based on traditional approaches and paradigms of nonlinear dynamics including dynamical chaos. Neural systems are, however, an unusual subject for nonlinear dynamics for several reasons: (i) Even the simplest neural network, with only a few neurons and synaptic connections, has an enormous number of variables and control parameters. These make neural systems adaptive and flexible, and are critical to their biological function. (ii) In contrast to traditional physical systems described by well-known basic principles, first principles governing the dynamics of neural systems are unknown. (iii) Many different neural systems exhibit similar dynamics despite having different architectures and different levels of complexity. (iv) The network architecture and connection strengths are usually not known in detail and therefore the dynamical analysis must, in some sense, be probabilistic. (v) Since nervous systems are able to organize behavior based on sensory inputs, the dynamical modeling of these systems has to explain the transformation of temporal information into combinatorial or combinatorial-temporal codes, and vice versa, for memory and recognition. In this review these problems are discussed in the context of addressing the stimulating questions: What can neuroscience learn from nonlinear dynamics, and what can nonlinear dynamics learn from neuroscience?This work was supported by NSF Grant No. NSF/EIA-0130708, and Grant No. PHY 0414174; NIH Grant No. 1 R01 NS50945 and Grant No. NS40110; MEC BFI2003-07276, and Fundación BBVA

2016 Summer Research Symposium Abstract Book

2016 Summer volume of abstracts for science research projects conducted by students at Trinity College