Phase response theory explains cluster formation in sparsely but strongly connected inhibitory neural networks and effects of jitter due to sparse connectivity

Existence and Stability Criteria for Global Synchrony and for Synchrony in two Alternating Clusters of Pulse-Coupled Oscillators Updated to Include Conduction Delays

Phase Response Curves (PRCs) have been useful in determining and analyzing various phase-locking modes in networks of oscillators under pulse-coupling assumptions, as reviewed in Mathematical Biosciences, 226:77-96, 2010. Here, we update that review to include progress since 2010 on pulse coupled oscillators with conduction delays. We then present original results that extend the derivation of the criteria for stability of global synchrony in networks of pulse-coupled oscillators to include conduction delays. We also incorporate conduction delays to extend previous studies that showed how an alternating firing pattern between two synchronized clusters could enforce within cluster synchrony, even for clusters unable to synchronize themselves in isolation. To obtain these results, we used self-connected neurons to represent clusters. These results greatly extend the applicability of the stability analyses to networks of pulse-coupled oscillators since conduction delays are ubiquitous and strongly impact the stability of synchrony. Although these analyses only strictly apply to identical oscillators with identical connections to other oscillators, the principles are general and suggest how to promote or impede synchrony in physiological networks of neurons, for example. Heterogeneity can be interpreted as a form of frozen noise, and approximate synchrony can be sustained despite heterogeneity. The pulse-coupled oscillator model can not only be used to describe biological neuronal networks but also cardiac pacemakers, lasers, fireflies, artificial neural networks, social self-organization, and wireless sensor networks.

AMS Subject Classification

37N25, 39A06, 39A30, 92B25, 92C20.

How to entrain a selected neuronal rhythm but not others: open-loop dithered brain stimulation for selective entrainment

Objective.While brain stimulation therapies such as deep brain stimulation for Parkinson's disease (PD) can be effective, they have yet to reach their full potential across neurological disorders. Entraining neuronal rhythms using rhythmic brain stimulation has been suggested as a new therapeutic mechanism to restore neurotypical behaviour in conditions such as chronic pain, depression, and Alzheimer's disease. However, theoretical and experimental evidence indicate that brain stimulation can also entrain neuronal rhythms at sub- and super-harmonics, far from the stimulation frequency. Crucially, these counterintuitive effects could be harmful to patients, for example by triggering debilitating involuntary movements in PD. We therefore seek a principled approach to selectively promote rhythms close to the stimulation frequency, while avoiding potential harmful effects by preventing entrainment at sub- and super-harmonics.Approach.Our open-loop approach to selective entrainment, dithered stimulation, consists in adding white noise to the stimulation period.Main results.We theoretically establish the ability of dithered stimulation to selectively entrain a given brain rhythm, and verify its efficacy in simulations of uncoupled neural oscillators, and networks of coupled neural oscillators. Furthermore, we show that dithered stimulation can be implemented in neurostimulators with limited capabilities by toggling within a finite set of stimulation frequencies.Significance.Likely implementable across a variety of existing brain stimulation devices, dithering-based selective entrainment has potential to enable new brain stimulation therapies, as well as new neuroscientific research exploiting its ability to modulate higher-order entrainment.

Interneuronal network model of theta-nested fast oscillations predicts differential effects of heterogeneity, gap junctions and short term depression for hyperpolarizing versus shunting inhibition

Theta and gamma oscillations in the hippocampus have been hypothesized to play a role in the encoding and retrieval of memories. Recently, it was shown that an intrinsic fast gamma mechanism in medial entorhinal cortex can be recruited by optogenetic stimulation at theta frequencies, which can persist with fast excitatory synaptic transmission blocked, suggesting a contribution of interneuronal network gamma (ING). We calibrated the passive and active properties of a 100-neuron model network to capture the range of passive properties and frequency/current relationships of experimentally recorded PV+ neurons in the medial entorhinal cortex (mEC). The strength and probabilities of chemical and electrical synapses were also calibrated using paired recordings, as were the kinetics and short-term depression (STD) of the chemical synapses. Gap junctions that contribute a noticeable fraction of the input resistance were required for synchrony with hyperpolarizing inhibition; these networks exhibited theta-nested high frequency oscillations similar to the putative ING observed experimentally in the optogenetically-driven PV-ChR2 mice. With STD included in the model, the network desynchronized at frequencies above ~200 Hz, so for sufficiently strong drive, fast oscillations were only observed before the peak of the theta. Because hyperpolarizing synapses provide a synchronizing drive that contributes to robustness in the presence of heterogeneity, synchronization decreases as the hyperpolarizing inhibition becomes weaker. In contrast, networks with shunting inhibition required non-physiological levels of gap junctions to synchronize using conduction delays within the measured range.

Local inhibition in a model of the indirect pathway globus pallidus network slows and deregularizes background firing, but sharpens and synchronizes responses to striatal input

The external segment of globus pallidus (GPe) is a network of oscillatory neurons connected by inhibitory synapses. We studied the intrinsic dynamic and the response to a shared brief inhibitory stimulus in a model GPe network. Individual neurons were simulated using a phase resetting model based on measurements from mouse GPe neurons studied in slices. The neurons showed a broad heterogeneity in their firing rates and in the shapes and sizes of their phase resetting curves. Connectivity in the network was set to match experimental measurements. We generated statistically equivalent neuron heterogeneity in a small-world model, in which 99% of connections were made with near neighbors and 1% at random, and in a model with entirely random connectivity. In both networks, the resting activity was slowed and made more irregular by the local inhibition, but it did not show any periodic pattern. Cross-correlations among neuron pairs were limited to directly connected neurons. When stimulated by a shared inhibitory input, the individual neuron responses separated into two groups: one with a short and stereotyped period of inhibition followed by a transient increase in firing probability, and the other responding with a sustained inhibition. Despite differences in firing rate, the responses of the first group of neurons were of fixed duration and were synchronized across cells.

Polynomial, piecewise-Linear, Step (PLS): A Simple, Scalable, and Efficient Framework for Modeling Neurons

Biological neurons can be modeled with different levels of biophysical/biochemical details. The accuracy with which a model reflects the actual physiological processes and ultimately the information function of a neuron, can range from very detailed to a schematic phenomenological representation. This range exists due to the common problem: one needs to find an optimal trade-off between the level of details needed to capture the necessary information processing in a neuron and the computational load needed to compute 1 s of model time. An increase in modeled network size or model-time, for which the solution should be obtained, makes this trade-off pivotal in model development. Numerical simulations become incredibly challenging when an extensive network with a detailed representation of each neuron needs to be modeled over a long time interval to study slow evolving processes, e.g., development of the thalamocortical circuits. Here we suggest a simple, powerful and flexible approach in which we approximate the right-hand sides of differential equations by combinations of functions from three families: Polynomial, piecewise-Linear, Step (PLS). To obtain a single coherent framework, we provide four core principles in which PLS functions should be combined. We show the rationale behind each of the core principles. Two examples illustrate how to build a conductance-based or phenomenological model using the PLS-framework. We use the first example as a benchmark on three different computational platforms: CPU, GPU, and mobile system-on-chip devices. We show that the PLS-framework speeds up computations without increasing the memory footprint and maintains high model fidelity comparable to the fully-computed model or with lookup-table approximation. We are convinced that the full range of neuron models: from biophysical to phenomenological and even to abstract models, may benefit from using the PLS-framework.

Analyzing synchronized clusters in neuron networks

The presence of synchronized clusters in neuron networks is a hallmark of information transmission and processing. Common approaches to study cluster synchronization in networks of coupled oscillators ground on simplifying assumptions, which often neglect key biological features of neuron networks. Here we propose a general framework to study presence and stability of synchronous clusters in more realistic models of neuron networks, characterized by the presence of delays, different kinds of neurons and synapses. Application of this framework to two examples with different size and features (the directed network of the macaque cerebral cortex and the swim central pattern generator of a mollusc) provides an interpretation key to explain known functional mechanisms emerging from the combination of anatomy and neuron dynamics. The cluster synchronization analysis is carried out also by changing parameters and studying bifurcations. Despite some modeling simplifications in one of the examples, the obtained results are in good agreement with previously reported biological data.

Shunting Inhibition Improves Synchronization in Heterogeneous Inhibitory Interneuronal Networks with Type 1 Excitability Whereas Hyperpolarizing Inhibition Is Better for Type 2 Excitability

All-to-all homogeneous networks of inhibitory neurons synchronize completely under the right conditions; however, many modeling studies have shown that biological levels of heterogeneity disrupt synchrony. Our fundamental scientific question is “how can neurons maintain partial synchrony in the presence of heterogeneity and noise?” A particular subset of strongly interconnected interneurons, the PV+ fast-spiking (FS) basket neurons, are strongly implicated in γ oscillations and in phase locking of nested γ oscillations to theta. Their excitability type apparently varies between brain regions: in CA1 and the dentate gyrus they have type 1 excitability, meaning that they can fire arbitrarily slowly, whereas in the striatum and cortex they have type 2 excitability, meaning that there is a frequency thresh old below which they cannot sustain repetitive firing. We constrained the models to study the effect of excitability type (more precisely bifurcation type) in isolation from all other factors. We use sparsely connected, heterogeneous, noisy networks with synaptic delays to show that synchronization properties, namely the resistance to suppression and the strength of theta phase to γ amplitude coupling, are strongly dependent on the pairing of excitability type with the type of inhibition. Shunting inhibition performs better for type 1 and hyperpolarizing inhibition for type 2. γ Oscillations and their nesting within theta oscillations are thought to subserve cognitive functions like memory encoding and recall; therefore, it is important to understand the contribution of intrinsic properties to these rhythms.

Inhibitory Network Bistability Explains Increased Interneuronal Activity Prior to Seizure Onset

Recent experimental literature has revealed that GABAergic interneurons exhibit increased activity prior to seizure onset, alongside additional evidence that such activity is synchronous and may arise abruptly. These findings have led some to hypothesize that this interneuronal activity may serve a causal role in driving the sudden change in brain activity that heralds seizure onset. However, the mechanisms predisposing an inhibitory network toward increased activity, specifically prior to ictogenesis, without a permanent change to inputs to the system remain unknown. We address this question by comparing simulated inhibitory networks containing control interneurons and networks containing hyperexcitable interneurons modeled to mimic treatment with 4-Aminopyridine (4-AP), an agent commonly used to model seizures in vivo and in vitro. Our in silico study demonstrates that model inhibitory networks with 4-AP interneurons are more prone than their control counterparts to exist in a bistable state in which asynchronously firing networks can abruptly transition into synchrony driven by a brief perturbation. This transition into synchrony brings about a corresponding increase in overall firing rate. We further show that perturbations driving this transition could arise in vivo from background excitatory synaptic activity in the cortex. Thus, we propose that bistability explains the increase in interneuron activity observed experimentally prior to seizure via a transition from incoherent to coherent dynamics. Moreover, bistability explains why inhibitory networks containing hyperexcitable interneurons are more vulnerable to this change in dynamics, and how such networks can undergo a transition without a permanent change in the drive. We note that while our comparisons are between networks of control and ictogenic neurons, the conclusions drawn specifically relate to the unusual dynamics that arise prior to seizure, and not seizure onset itself. However, providing a mechanistic explanation for this phenomenon specifically in a pro-ictogenic setting generates experimentally testable hypotheses regarding the role of inhibitory neurons in pre-ictal neural dynamics, and motivates further computational research into mechanisms underlying a newly hypothesized multi-step pathway to seizure initiated by inhibition.