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

Desynchronization of large-scale neural networks by stabilizing unknown unstable incoherent equilibrium states

In large-scale neural networks, coherent limit cycle oscillations usually coexist with unstable incoherent equilibrium states, which are not observed experimentally. We implement a first-order dynamic controller to stabilize unknown equilibrium states and suppress coherent oscillations. The stabilization of incoherent equilibria associated with unstable focus and saddle is considered. The algorithm is demonstrated for networks composed of quadratic integrate-and-fire (QIF) neurons and Hindmarsh-Rose neurons. The microscopic equations of an infinitely large QIF neural network can be reduced to an exact low-dimensional system of mean-field equations, which makes it possible to study the control problem analytically.Comment: 11 pages, 9 figure

Sparse Gamma Rhythms Arising through Clustering in Adapting Neuronal Networks

Gamma rhythms (30–100 Hz) are an extensively studied synchronous brain state responsible for a number of sensory, memory, and motor processes. Experimental evidence suggests that fast-spiking interneurons are responsible for carrying the high frequency components of the rhythm, while regular-spiking pyramidal neurons fire sparsely. We propose that a combination of spike frequency adaptation and global inhibition may be responsible for this behavior. Excitatory neurons form several clusters that fire every few cycles of the fast oscillation. This is first shown in a detailed biophysical network model and then analyzed thoroughly in an idealized model. We exploit the fact that the timescale of adaptation is much slower than that of the other variables. Singular perturbation theory is used to derive an approximate periodic solution for a single spiking unit. This is then used to predict the relationship between the number of clusters arising spontaneously in the network as it relates to the adaptation time constant. We compare this to a complementary analysis that employs a weak coupling assumption to predict the first Fourier mode to destabilize from the incoherent state of an associated phase model as the external noise is reduced. Both approaches predict the same scaling of cluster number with respect to the adaptation time constant, which is corroborated in numerical simulations of the full system. Thus, we develop several testable predictions regarding the formation and characteristics of gamma rhythms with sparsely firing excitatory neurons

Stochastic Resonance Modulates Neural Synchronization within and between Cortical Sources

Neural synchronization is a mechanism whereby functionally specific brain regions establish transient networks for perception, cognition, and action. Direct addition of weak noise (fast random fluctuations) to various neural systems enhances synchronization through the mechanism of stochastic resonance (SR). Moreover, SR also occurs in human perception, cognition, and action. Perception, cognition, and action are closely correlated with, and may depend upon, synchronized oscillations within specialized brain networks. We tested the hypothesis that SR-mediated neural synchronization occurs within and between functionally relevant brain areas and thus could be responsible for behavioral SR. We measured the 40-Hz transient response of the human auditory cortex to brief pure tones. This response arises when the ongoing, random-phase, 40-Hz activity of a group of tuned neurons in the auditory cortex becomes synchronized in response to the onset of an above-threshold sound at its “preferred” frequency. We presented a stream of near-threshold standard sounds in various levels of added broadband noise and measured subjects' 40-Hz response to the standards in a deviant-detection paradigm using high-density EEG. We used independent component analysis and dipole fitting to locate neural sources of the 40-Hz response in bilateral auditory cortex, left posterior cingulate cortex and left superior frontal gyrus. We found that added noise enhanced the 40-Hz response in all these areas. Moreover, added noise also increased the synchronization between these regions in alpha and gamma frequency bands both during and after the 40-Hz response. Our results demonstrate neural SR in several functionally specific brain regions, including areas not traditionally thought to contribute to the auditory 40-Hz transient response. In addition, we demonstrated SR in the synchronization between these brain regions. Thus, both intra- and inter-regional synchronization of neural activity are facilitated by the addition of moderate amounts of random noise. Because the noise levels in the brain fluctuate with arousal system activity, particularly across sleep-wake cycles, optimal neural noise levels, and thus SR, could be involved in optimizing the formation of task-relevant brain networks at several scales under normal conditions

Restoration of rhythmicity in diffusively coupled dynamical networks

We acknowledge financial support from the National Natural Science Foundation of China (No. 11202082, No. 61203235, No. 11371367 and No. 11271290), the Fundamental Research Funds for the Central Universities of China under Grant No. 2014QT005, IRTG1740(DFG-FAPESP), and SERB-DST Fast Track scheme for young scientist under Grant No. ST/FTP/PS-119/2013, NSF CHE-0955555 and Grant No. 229171/2013-3 (CNPq).Peer reviewedPublisher PD

Synchrony-induced modes of oscillation of a neural field model

We investigate the modes of oscillation of heterogeneous ring-networks of quadratic integrate-and-fire (QIF) neurons with non-local, space-dependent coupling. Perturbations of the equilibrium state with a particular wave number produce transient standing waves with a specific temporal frequency, analogous to those in a tense string. In the neuronal network, the equilibrium corresponds to a spatially homogeneous, asynchronous state. Perturbations of this state excite the network’s oscillatory modes, which reflect the interplay of episodes of synchronous spiking with the excitatory-inhibitory spatial interactions. In the thermodynamic limit, an exact low-dimensional neural field model (QIF-NFM) describing the macroscopic dynamics of the network is derived. This allows us to obtain formulas for the Turing eigenvalues of the spatially-homogeneous state, and hence to obtain its stability boundary. We find that the frequency of each Turing mode depends on the corresponding Fourier coefficient of the synaptic pattern of connectivity. The decay rate instead, is identical for all oscillation modes as a consequence of the heterogeneity-induced desynchronization of the neurons. Finally, we numerically compute the spectrum of spatially-inhomogeneous solutions branching from the Turing bifurcation, showing that similar oscillatory modes operate in neural bump states, and are maintained away from onset

The response of a classical Hodgkin–Huxley neuron to an inhibitory input pulse

A population of uncoupled neurons can often be brought close to synchrony by a single strong inhibitory input pulse affecting all neurons equally. This mechanism is thought to underlie some brain rhythms, in particular gamma frequency (30–80 Hz) oscillations in the hippocampus and neocortex. Here we show that synchronization by an inhibitory input pulse often fails for populations of classical Hodgkin–Huxley neurons. Our reasoning suggests that in general, synchronization by inhibitory input pulses can fail when the transition of the target neurons from rest to spiking involves a Hopf bifurcation, especially when inhibition is shunting, not hyperpolarizing. Surprisingly, synchronization is more likely to fail when the inhibitory pulse is stronger or longer-lasting. These findings have potential implications for the question which neurons participate in brain rhythms, in particular in gamma oscillations

Dynamic Control of Network Level Information Processing through Cholinergic Modulation

Acetylcholine (ACh) release is a prominent neurochemical marker of arousal state within the brain. Changes in ACh are associated with changes in neural activity and information processing, though its exact role and the mechanisms through which it acts are unknown. Here I show that the dynamic changes in ACh levels that are associated with arousal state control informational processing functions of networks through its effects on the degree of Spike-Frequency Adaptation (SFA), an activity dependent decrease in excitability, synchronizability, and neuronal resonance displayed by single cells. Using numerical modeling I develop mechanistic explanations for how control of these properties shift network activity from a stable high frequency spiking pattern to a traveling wave of activity. This transition mimics the change in brain dynamics seen between high ACh states, such as waking and Rapid Eye Movement (REM) sleep, and low ACh states such as Non-REM (NREM) sleep. A corresponding, and related, transition in network level memory recall is also occurs as ACh modulates neuronal SFA. When ACh is at its highest levels (waking) all memories are stably recalled, as ACh is decreased (REM) in the model weakly encoded memories destabilize while strong memories remain stable. In levels of ACh that match Slow Wave Sleep (SWS), no encoded memories are stably recalled. This results from a competition between SFA and excitatory input strength and provides a mechanism for neural networks to control the representation of underlying synaptic information. Finally I show that during the low ACh conditions, oscillatory conditions allow for external inputs to be properly stored in and recalled from synaptic weights. Taken together this work demonstrates that dynamic neuromodulation is critical for the regulation of information processing tasks in neural networks. These results suggest that ACh is capable of switching networks between two distinct information processing modes. Rate coding of information is facilitated during high ACh conditions and phase coding of information is facilitated during low ACh conditions. Finally I propose that ACh levels control whether a network is in one of three functional states: (High ACh; Active waking) optimized for encoding of new information or the stable representation of relevant memories, (Mid ACh; resting state or REM) optimized for encoding connections between currently stored memories or searching the catalog of stored memories, and (Low ACh; NREM) optimized for renormalization of synaptic strength and memory consolidation. This work provides a mechanistic insight into the role of dynamic changes in ACh levels for the encoding, consolidation, and maintenance of memories within the brain.PHDNeuroscienceUniversity of Michigan, Horace H. Rackham School of Graduate Studieshttps://deepblue.lib.umich.edu/bitstream/2027.42/147503/1/roachjp_1.pd