Mathematical neuroscience

Monday, June 14 at 09:30am (PDT)
Monday, June 14 at 05:30pm (BST)
Tuesday, June 15 01:30am (KST)

SMB2021 SMB2021 Follow Monday (Tuesday) during the "MS01" time block.
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Sunil Modhara (University of Nottingham, United Kingdom), Stephen Coombes (University of Nottingham, United Kingdom), Ruediger Thul (University of Nottingham, United Kingdom), Daniele Avitabile (Vrije Universiteit Amsterdam, Netherlands)


This session approaches mathematical neuroscience from a number of different perspectives, bringing together clinically motivated topics, neural field-type models, and a model of neuronal beat generation. Common to all talks is the idea of using non-linear dynamical systems techniques to further our understanding of the brain. From individual neurons and networks, to sub-structures such as the thalamus, and finally to the brain as a whole, neural modelling is inherently a multiscale discipline. The models to be discussed in this mini- symposium exemplify this broad spectrum, with single-cell interactions at the synapse, single beat generator neurons, and mutations of ion channels at smaller spatial scales, to mean field models at the larger spatial scale. Topics considered in this session include rhythmogenesis and entrainment, conductance-based modelling, non-smooth dynamical systems, learning, synchronisation, bistable systems, and bifurcation analysis.

Sunil Modhara

(University of Nottingham, United Kingdom)
"Neural fields with rebound currents: novel routes to patterning"
The understanding of how spatio-temporal patterns of neural activity may arise in the cortex of the brain has advanced with the development and analysis of neural field models. To replicate this success for sub-cortical tissues, such as the thalamus, requires an extension to include relevant ionic currents that can further shape firing response. Here we advocate for one such approach that can accommodate slow currents. By way of illustration we focus on incorporating a T-type calcium current into the standard neural field framework. Direct numerical simulations are used to show that the resulting tissue model has many of the properties seen in more biophysically detailed model studies, and most importantly the generation of oscillations, waves, and patterns that arise from rebound firing. To explore the emergence of such solutions we focus on one- and two-dimensional spatial models and show that exact solutions describing homogeneous oscillations can be constructed in the limit that the firing rate nonlinearity is a Heaviside function. A linear stability analysis, using techniques from non-smooth dynamical systems, is used to determine the points at which bifurcations from synchrony can occur. Furthermore, we construct periodic travelling waves and investigate their stability with the use of an appropriate Evans function. The stable branches of the dispersion curve for periodic travelling waves are found to be in excellent agreement with simulations initiated from an unstable branch of the synchronous solution.

Louisiane Lemaire

(Inria Sophia Antipolis Méditerranée Research Centre, France)
"Mathematical model of the mutations of a sodium channel (NaV1.1) capturing both migraine and epilepsy scenarios"
NaV1.1 is a Na+ voltage-gated channel expressed in GABAergic neurons, crucial for their excitability. Gain-of-function mutations of this channel cause familial hemiplegic migraine type 3 (FHM3), while loss-of-function mutations lead to epilepsy. The pathological mechanisms are unclear. Cortical spreading depression (CSD) is a wave of intense firing followed by depolarization block, propagating in the cortex. In the case of FHM3, it is thought that CSD sensitizes meningeal nociceptors, inducing headache. However, the link between FHM3 mutations and the initiation of CSD remains to be understood. We develop a two-neuron (GABAergic and pyramidal) conductance-based model with dynamic ion concentrations, since the considered pathologies disrupt ionic gradients. Keeping the other parameter values unchanged, we implement FMH3 mutations using persistent Na+ current and epilepsy ones with reduced transient Na+ current. Our results suggest the importance of other mechanisms of action of NaV1.1 mutations than changes in the firing frequency of GABAergic neurons. In our model, FHM3 mutations modify ion fluxes at each action potential. The resulting accumulation of extracellular potassium facilitates CSD initiation and reduces its latency, in agreement with recent experimental findings. Epilepsy mutations make the GABAergic neuron more susceptible to depolarization block. The removal of their inhibitory restraint causes a simultaneous increase of the pyramidal cell's firing frequency.

Manu Kalia

(University of Twente, Netherlands)
"Modeling ischemic vulnerability at the tripartite synapse"
The work that I will present is about single-cell neuron-astrocyte interactions at the synapse. The model considered describes ion dynamics at such an interaction during ischemia (which means low oxygen due to reduced blood flow, say, during the onset of stroke). I will present some results about bistability in the model and the tipping from one state to another, and some predictions about possible pharmacological blockers to recover cells from ischemic injury.

John Rinzel

(New York University, USA)
"A neuronal model for learning to keep a rhythmic beat"
When listening to music, we typically lock onto and move to a beat (1-6 Hz). Behavioral studies on such synchronization (Repp 2005) abound, yet the neural mechanisms remain poorly understood. Some models hypothesize an array of self-sustaining entrainable neural oscillators that resonate when forced with rhythmic stimuli (Large et al. 2010). In contrast, our formulation focuses on event time estimation and plasticity: a neuronal beat generator that adapts its intrinsic frequency and phase to match the extermal rhythm. The model quickly learns new rhythms, within a few cycles as found in human behavior. When the stimulus is removed the beat generator continues to produce the learned rhythm in accordance with a synchronization continuation task.

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Virtual conference of the Society for Mathematical Biology, 2021.