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CAMBAM Zoomposium: Multiple Timescales in Neuronal and Other Systems

5 juin 2020

Cohosted by the University of Waterloo and the Fields Institute


The generation of neuronal activity – from spikes, to bursts, to multi-phase rhythms – fundamentally involves the interaction of processes that evolve on widely disparate timescales. As a result, advances in theoretical neuroscience and in methods for the analysis of multiple timescale dynamics have emerged synergistically, with experimental observations driving theoretical developments and with theoretical advances yielding new explanations for data. In this session, speakers will present work featuring advances on both sides of this partnership, which highlights new findings about neuronal and other biological systems together with the modern approaches to multiple timescale analysis that underlie these results.

To see the lecture’s video recordings

First Session (10:00 – 11:30 AM - EDT)

10:00 – 10:30 --- Krasimira Tsaneva-Atanasova (Professor of Mathematics for Healthcare, University of Exeter)

Title: Pseudo-Plateau Bursting and Mixed-Mode Oscillations in a Model of Developing Inner Hair Cells


Inner hair cells (IHCs) are excitable sensory cells in the inner ear that encode acoustic information. Before the onset of hearing IHCs fire calcium-based action potentials that trigger transmitter release onto developing spiral ganglion neurones. There is accumulating experimental evidence that these spontaneous firing patterns are associated with maturation of the IHC synapses and hence involved in the development of hearing. Building on our previous modelling work we propose a three-dimensional, reduced IHC model and carry out non-dimensionalisation. We show that there is a significant range of parameter values for which the dynamics of the reduced (three-dimensional) model map well onto the dynamics observed in the original biophysical (four-dimensional) IHC model. By estimating the typical time scales in the reduced IHC model we demonstrate that this model could be characterised by two fast and one slow or one fast and two slow variables. We investigate how changes in the conductance of the voltage-gated calcium channels as well as the fraction of free cytosolic calcium concentration in the model affect the oscillatory model bahaviour leading to transition from pseudo-plateau bursting to mixed-mode oscillations. Hence, using fast-slow analysis we are able to further our understanding of this model and reveal a path in the parameter space connecting pseudo-plateau bursting and mixed-mode oscillations by varying a single parameter in the model.

10:30 – 11:00 --- Elif Koksal Ersoz (Postdoc Researcher, Inserm)

Title: Neural Mass Modeling of Slow-Fast Dynamics of Seizure Initiation and Abortion


Epilepsies refer to a neurological disorder affecting about 1% of the worldwide population. They are characterized by recurrent seizures that consist of episodes of paroxysmal neural discharges. During seizures, specific brain activity (rhythmic spikes, fast onset) is observed in electrophysiological data, typically in local field potentials (LFPs) recorded by SEEG electrodes. Neurophysiologically-plausible lumped-parameter models at the mesoscopic scale, namely neural mass models (NMMs), describe the average activity of neural subpopulations of main cells and interneurons. They are widely accepted as computational models of epilepsy. Notably, the Wendling-Chauvel NMM (Wendling et al., 2002) has been shown to not only mimic the epileptic LFP signals but also give insights on the physiological mechanisms related to different stages of a seizure due to presence of a two types of inhibitory interneurons with different synaptic time constants. In the former studies on the bifurcation analysis of the model have provided a dictionary of cortical patterns of activity and linked the seizure initiation and termination to a slow-fast process. Here, we consider the Wendling-Chauvel NMM. We investigate how the multiple timescale dynamics and the inhibition/excitation ratio shape together the epileptic discharges. We show that electrical pulses perturbing the SOM+ interneurons can induce a switch form epileptic to background activity.

11:00 – 11:30 --- Jonathan Touboul (Associate Professor of Mathematics, Brandeis University)

Title: Slow Chaos, Fast Chaos and Homeostasis in Neuronal Activity


A number of natural phenomena and models qualify as chaotic mathematically, yet their dynamics are show a certain degree of regularity which may be sufficient to maintain appropriate function. For instance, the stomatogastric ganglion of crabs generates relatively slow periodic activity with 3 populations activating sequentially, but where neurons activate in chaotic bursts during their activation phase. These networks ensure appropriate function until excessive perturbations of the environment (e.g., temperature) leads populations start activating in a disorderly manner preventing the system from maintaining its function (see Marder Alonso, eLife 2019).

We will investigate, define and study these two types of chaos in the context of slow-fast dynamical systems with chaos in the fast variable. I will show a transition between 'fast chaos' trajectories (where the slow dynamics remains approximately regular), and `slow chaos' whereby slow dynamics is strongly affected by chaos on the fast variable. We will provide general conditions for a system to be in one type of chaos or the other, and study the transition between slow- and fast-chaos. I will illustrate these concepts on the Rulkov neuron model (possibly the simplest model supporting slow-fast chaos) and a simplified model of the crab stomatogastric ganglion.

Second Session (12:30 – 2:00 PM - EDT)

12:30 – 1:00 --- Saeed Farjami (Postdoctoral Research Associate, University of Surrey)

Title: Bursting in Cerebellar Stellate Cells Induced by Pharmacological Agents: Non-Sequential Spike Adding


Cerebellar stellate cells (CSC) are inhibitory interneurons that synapse onto Purkinje cells. Our group has extensively studied their electrophysiological properties, including switching in responsiveness, the non-monotonic first-spike latency and run-up. Recent experimental evidence has shown that these neurons also possess the machinery to burst when treated with certain pharmacological agents separately or jointly. Indeed, treatment with 4AP, a partial blocker of delayed rectifier and A-type K+ channels, induced a bursting profile in CSCs significantly different than that produced when 4AP is applied in combination with Cd2+, a blocker of high voltage activated (HVA) Ca2+ channels. By extending an HH-model we previously revised to include HVA and Ca2+-activated K+ (KCa) channels, we showed that the model preserves its previous properties while simultaneously explaining how 4AP and Cd2+ induce the two burst profiles. We demonstrated that 4AP is likely potentiating HVA while Cd2+ is potentiating KCa. Our slow fast analysis revealed how these bursts are generated and showed that spike-adding in 4AP-generated bursts is non-sequential when changing HVA and KCa conductances, a feature never observed in other HH-type models. It also explained the role of delayed Hopf in generating non-spiking plateau in the active phase of 4AP+Cd2+-generated bursts. In this talk, we will provide an overview of the results listed above.

1:00 – 1:30 --- Yangyang Wang (Assistant Professor of Mathematics, University of Iowa)

Title: Complex Bursting Patterns in an Embryonic Respiratory Neuron Model


Pre-Bötzinger complex (pre-BötC) network activity within the mammalian brainstem controls the inspiratory phase of the respiratory rhythm. While bursting in pre-BötC neurons during the postnatal period has been extensively studied, less is known regarding inspiratory pacemaker neuron behavior at embryonic stages. Recent data in mouse embryo brainstem slices have revealed the existence of a variety of bursting activity patterns depending on distinct combinations of burst-generating INaP and ICAN conductances. In this work, we consider a model of an isolated embryonic pre-BötC neuron featuring two distinct bursting mechanisms. We use methods of dynamical systems theory, such as phase plane analysis, fast-slow decomposition, and bifurcation analysis, to uncover mechanisms underlying several different types of intrinsic bursting dynamics observed experimentally including several forms of plateau bursts, bursts involving depolarization block, and various combinations of these patterns. Our analysis also yields predictions about how changes in the balance of the two bursting mechanisms contribute to alterations in inspiratory pacemaker neuron activity during prenatal development.

1:30 – 2:00 --- Marcello Codianni (PhD Student, University of Pittsburgh)

Title: Limb Segment Control in Stickbug Locomotion


We investigate a neuromechanical mechanism for coordinating control of the protrator-retractor limb segment in the stick insect in a tetrapodal gait pattern. In this system, sensory feedback from the distal limb segent is modeled as periodic bottom-up forcing onto a system of excitable interneurons. These interneurons then feed this signal forward onto a set of half-center oscillators, which are also modulated by top-down signaling. In this work we construct parameter regions where this protractor-retractor network can entrain to bottom-up signals to effectively maintain gait and speed. When entrainment is lost, it can be recovered through modulation of top-down signals. We also characterize the possibility and maintenance of entrainment in the case of damaged or destroyed connections between the interneurons and the half-center oscillator. Finally, we investigate the ability of top-down signaling to modify movement timing, allowing for gait shifts and altered phase relationships.

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