Shared inputs, entrainment, and desynchrony in elliptic bursters: from slow passage to discontinuous circle maps

What input signals will lead to synchrony vs. desynchrony in a group of biological oscillators? This question connects with both classical dynamical systems analyses of entrainment and phase locking and with emerging studies of stimulation patterns for controlling neural network activity. Here, we focus on the response of a population of uncoupled, elliptically bursting neurons to a common pulsatile input. We extend a phase reduction from the literature to capture inputs of varied strength, leading to a circle map with discontinuities of various orders. In a combined analytical and numerical approach, we apply our results to both a normal form model for elliptic bursting and to a biophysically-based neuron model from the basal ganglia. We find that, depending on the period and amplitude of inputs, the response can either appear chaotic (with provably positive Lyaponov exponent for the associated circle maps), or periodic with a broad range of phase-locked periods. Throughout, we discuss the critical underlying mechanisms, including slow-passage effects through Hopf bifurcation, the role and origin of discontinuities, and the impact of noiseComment: 17 figures, 40 page

Huygens synchronisation of three clocks equidistant from each other

In this paper we study the synchronisation of three identical oscillators, i.e., clocks, hanging from the same hard support. We consider the case where each clock interacts with the other two clocks. The synchronisation is attained through the exchange of small impacts between each pair of oscillators. The fundamental result of this article is that the final locked state is at phase difference of ((2{\pi})/3) from successive clocks (clockwise or counter-clockwise). Moreover, the locked states attract a set whose closure is the global set of initial conditions. The methodology of our analysis consists in the construction a model, which is a non-linear discrete dynamical system, i.e. a non-linear difference equation. The results are extendable to any set of three oscillators under mutual symmetric interaction, despite the particular models of the oscillators

Fast Synchronization of Perpetual Grouping in Laminar Visual Cortical Circuits

Perceptual grouping is well-known to be a fundamental process during visual perception, notably grouping across scenic regions that do not receive contrastive visual inputs. Illusory contours are a classical example of such groupings. Recent psychophysical and neurophysiological evidence have shown that the grouping process can facilitate rapid synchronization of the cells that are bound together by a grouping, even when the grouping must be completed across regions that receive no contrastive inputs. Synchronous grouping can hereby bind together different object parts that may have become desynchronized due to a variety of factors, and can enhance the efficiency of cortical transmission. Neural models of perceptual grouping have clarified how such fast synchronization may occur by using bipole grouping cells, whose predicted properties have been supported by psychophysical, anatomical, and neurophysiological experiments. These models have not, however, incorporated some of the realistic constraints on which groupings in the brain are conditioned, notably the measured spatial extent of long-range interactions in layer 2/3 of a grouping network, and realistic synaptic and axonal signaling delays within and across cells in different cortical layers. This work addresses the question: Can long-range interactions that obey the bipole constraint achieve fast synchronization under realistic anatomical and neurophysiological constraints that initially desynchronize grouping signals? Can the cells that synchronize retain their analog sensitivity to changing input amplitudes? Can the grouping process complete and synchronize illusory contours across gaps in bottom-up inputs? Our simulations show that the answer to these questions is Yes.Office of Naval Research (N00014-01-1-0624); Air Force Office of Scientific Research (F49620-01-1-03097

Macroscopic Models and Phase Resetting of Coupled Biological Oscillators

This thesis concerns the derivation and analysis of macroscopic mathematical models for coupled biological oscillators. Circadian rhythms, heart beats, and brain waves are all examples of biological rhythms formed through the aggregation of the rhythmic contributions of thousands of cellular oscillations. These systems evolve in an extremely high-dimensional phase space having at least as many degrees of freedom as the number of oscillators. This high-dimensionality often contrasts with the low-dimensional behavior observed on the collective or macroscopic scale. Moreover, the macroscopic dynamics are often of greater interest in biological applications. Therefore, it is imperative that mathematical techniques are developed to extract low-dimensional models for the macroscopic behavior of these systems. One such mathematical technique is the Ott-Antonsen ansatz. The Ott-Antonsen ansatz may be applied to high-dimensional systems of heterogeneous coupled oscillators to derive an exact low-dimensional description of the system in terms of macroscopic variables. We apply the Ott-Antonsen technique to determine the sensitivity of collective oscillations to perturbations with applications to neuroscience. The power of the Ott-Antonsen technique comes at the expense of several limitations which could limit its applicability to biological systems. To address this we compare the Ott-Antonsen ansatz with experimental measurements of circadian rhythms and numerical simulations of several other biological systems. This analysis reveals that a key assumption of the Ott-Antonsen approach is violated in these systems. However, we discover a low-dimensional structure in these data sets and characterize its emergence through a simple argument depending only on general phase-locking behavior in coupled oscillator systems. We further demonstrate the structure's emergence in networks of noisy heterogeneous oscillators with complex network connectivity. We show how this structure may be applied as an ansatz to derive low-dimensional macroscopic models for oscillator population activity. This approach allows for the incorporation of cellular-level experimental data into the macroscopic model whose parameters and variables can then be directly associated with tissue- or organism-level properties, thereby elucidating the core properties driving the collective behavior of the system. We first apply our ansatz to study the impact of light on the mammalian circadian system. To begin we derive a low-dimensional macroscopic model for the core circadian clock in mammals. Significantly, the variables and parameters in our model have physiological interpretations and may be compared with experimental results. We focus on the effect of four key factors which help shape the mammalian phase response to light: heterogeneity in the population of oscillators, the structure of the typical light phase response curve, the fraction of oscillators which receive direct light input and changes in the coupling strengths associated with seasonal day-lengths. We find these factors can explain several experimental results and provide insight into the processing of light information in the mammalian circadian system. In a second application of our ansatz we derive a pair of low-dimensional models for human circadian rhythms. We fit the model parameters to measurements of light sensitivity in human subjects, and validate these parameter fits with three additional data sets. We compare our model predictions with those made by previous phenomenological models for human circadian rhythms. We find our models make new predictions concerning the amplitude dynamics of the human circadian clock and the light entrainment properties of the clock. These results could have applications to the development of light-based therapies for circadian disorders.PHDApplied and Interdisciplinary MathematicsUniversity of Michigan, Horace H. Rackham School of Graduate Studieshttps://deepblue.lib.umich.edu/bitstream/2027.42/138766/1/khannay_1.pd

Metastability: an emergent phenomenon in networks of spiking neurons

It is widely recognised that different brain areas perform different specialised functions. However, it remains an open question how different brain areas coordinate with each other and give rise to global brain states and high-level cognition. Recent theories suggest that transient periods of synchronisation and desynchronisation provide a mechanism for dynamically integrating and forming coalitions of functionally related neural areas, and that at these times conditions are optimal for information transfer. Empirical evidence from human resting state networks has shown a tendency for multiple brain areas to synchronise for short amounts of time, and for different synchronous groups to appear at different times. In dynamical systems terms, this behaviour resembles metastability — an intrinsically driven movement between transient, attractor-like states. However, it remains an open question what the underlying mechanism is that gives rise to these observed phenomena. The thesis first establishes that oscillating neural populations display a great amount of spectral complexity, with several rhythms temporally coexisting in the same and different structures. The thesis next explores inter-band frequency modulation between neural oscillators. The results show that oscillations in different neural populations, and in different frequency bands, modulate each other so as to change frequency. Further to this, the interaction of these fluctuating frequencies in the network as a whole is able to drive different neural populations towards episodes of synchrony. Finally, a symbiotic relationship between metastability and underlying network structure is elucidated, in which the presence of plasticity, responding to the interactions between different neural areas, will naturally form modular small-world networks that in turn further promote metastability. This seemingly inevitable drive towards metastabilty in simulation suggests that it should also be present in biological brains. The conclusion drawn is that these key network characteristics, and the metastable dynamics they promote, facilitate versatile exploration, integration, and communication between functionally related neural areas, and thereby support sophisticated cognitive processing in the brain.Open Acces

LEGION-based image segmentation by means of spiking neural networks using normalized synaptic weights implemented on a compact scalable neuromorphic architecture

© . This manuscript version is made available under the CC-BY-NC-ND 4.0 license http://creativecommons.org/licenses/by-nc-nd/4.0/LEGION (Locally Excitatory, Globally Inhibitory Oscillator Network) topology has demonstrated good capabilities in scene segmentation applications. However, the implementation of LEGION algorithm requires machines with high performance to process a set of complex differential equations limiting its use in practical real-time applications. Recently, several authors have proposed alternative methods based on spiking neural networks (SNN) to create oscillatory neural networks with low computational complexity and highly feasible to be implemented on digital hardware to perform adaptive segmentation of images. Nevertheless, existing SNN with LEGION configuration focus on the membrane model leaving aside the behavior of the synapses although they play an important role in the synchronization of several segments by self-adapting their weights. In this work, we propose a SNN-LEGION configuration along with normalized weight of the synapses to self-adapt the SNN network to synchronize several segments of any size and shape at the same time. The proposed SNN-LEGION method involves a global inhibitor, which is in charge of performing the segmentation process between different objects with different sizes and shapes on time. To validate the proposal, the SNN-LEGION method is implemented on an optimized scalable neuromorphic architecture. Our preliminary results demonstrate that the proposed normalization process of the synaptic weights along with the SNN-LEGION configuration keep the capacity of the LEGION network to separate the segments on time, which can be useful in video processing applications such as vision processing systems for mobile robots, offering lower computational complexity and area consumption compared with previously reported solutions.The authors would like to thank the Consejo Nacional de Ciencia y Tecnologia (CONACyT) and the IPN for the financial support to realize this work under project SIP-20180251. This work was also supported in part by the Spanish Ministry of Science and Innovation and the European Social Fund (ESF) under Projects TEC2011-27047 and TEC2015-67278-R.Peer ReviewedPostprint (author's final draft