Average activity of excitatory and inhibitory neural populations

We develop an extension of the Ott-Antonsen method [E. Ott and T. M. Antonsen, Chaos 18(3), 037113 (2008)] that allows obtaining the mean activity (spiking rate) of a population of excitable units. By means of the Ott-Antonsen method, equations for the dynamics of the order parameters of coupled excitatory and inhibitory populations of excitable units are obtained, and their mean activities are computed. Two different excitable systems are studied: Adler units and theta neurons. The resulting bifurcation diagrams are compared with those obtained from studying the phenomenological Wilson-Cowan model in some regions of the parameter space. Compatible behaviors, as well as higher dimensional chaotic solutions, are observed. We study numerical simulations to further validate the equations.Fil: Roulet, Javier. Consejo Nacional de Investigaciones Científicas y Técnicas. Oficina de Coordinación Administrativa Ciudad Universitaria. Instituto de Física de Buenos Aires. Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales. Instituto de Física de Buenos Aires; ArgentinaFil: Mindlin, Bernardo Gabriel. Consejo Nacional de Investigaciones Científicas y Técnicas. Oficina de Coordinación Administrativa Ciudad Universitaria. Instituto de Física de Buenos Aires. Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales. Instituto de Física de Buenos Aires; Argentin

Feedback Control as a Framework for Understanding Tradeoffs in Biology

Control theory arose from a need to control synthetic systems. From regulating steam engines to tuning radios to devices capable of autonomous movement, it provided a formal mathematical basis for understanding the role of feedback in the stability (or change) of dynamical systems. It provides a framework for understanding any system with feedback regulation, including biological ones such as regulatory gene networks, cellular metabolic systems, sensorimotor dynamics of moving animals, and even ecological or evolutionary dynamics of organisms and populations. Here we focus on four case studies of the sensorimotor dynamics of animals, each of which involves the application of principles from control theory to probe stability and feedback in an organism's response to perturbations. We use examples from aquatic (electric fish station keeping and jamming avoidance), terrestrial (cockroach wall following) and aerial environments (flight control in moths) to highlight how one can use control theory to understand how feedback mechanisms interact with the physical dynamics of animals to determine their stability and response to sensory inputs and perturbations. Each case study is cast as a control problem with sensory input, neural processing, and motor dynamics, the output of which feeds back to the sensory inputs. Collectively, the interaction of these systems in a closed loop determines the behavior of the entire system.Comment: Submitted to Integr Comp Bio

Synchronization of electrically coupled resonate-and-fire neurons

Electrical coupling between neurons is broadly present across brain areas and is typically assumed to synchronize network activity. However, intrinsic properties of the coupled cells can complicate this simple picture. Many cell types with strong electrical coupling have been shown to exhibit resonant properties, and the subthreshold fluctuations arising from resonance are transmitted through electrical synapses in addition to action potentials. Using the theory of weakly coupled oscillators, we explore the effect of both subthreshold and spike-mediated coupling on synchrony in small networks of electrically coupled resonate-and-fire neurons, a hybrid neuron model with linear subthreshold dynamics and discrete post-spike reset. We calculate the phase response curve using an extension of the adjoint method that accounts for the discontinuity in the dynamics. We find that both spikes and resonant subthreshold fluctuations can jointly promote synchronization. The subthreshold contribution is strongest when the voltage exhibits a significant post-spike elevation in voltage, or plateau. Additionally, we show that the geometry of trajectories approaching the spiking threshold causes a "reset-induced shear" effect that can oppose synchrony in the presence of network asymmetry, despite having no effect on the phase-locking of symmetrically coupled pairs

Interacting Mechanisms Driving Synchrony in Neural Networks with Inhibitory Interneurons

Computational neuroscience contributes to our understanding of the brain by applying techniques from fields including mathematics, physics, and computer science to neuroscientific problems that are not amenable to purely biologic study. One area in which this interdisciplinary research is particularly valuable is the proposal and analysis of mechanisms underlying neural network behaviors. Neural synchrony, especially when driven by inhibitory interneurons, is a behavior of particular importance considering this behavior play a role in neural oscillations underlying important brain functions such as memory formation and attention. Typically, these oscillations arise from synchronous firing of a neural population, and thus the study of neural oscillations and neural synchrony are deeply intertwined. Such network behaviors are particularly amenable to computational analysis given the variety of mathematical techniques that are of use in this field. Inhibitory interneurons are thought to drive synchrony in ways described by two computational mechanisms: Interneuron Network Gamma (ING), which describes how an inhibitory network synchronizes itself; and Pyramidal Interneuron Network Gamma (PING), which describes how a population of interneurons inter-connected with a population of excitatory pyramidal cells (an E-I network) synchronizes both populations. As first articulated using simplified interneuron models, these mechanisms find network properties are the primary impetus for synchrony. However, as neurobiologists uncover interneurons exhibiting a vast array of cellular and intra-connectivity properties, our understanding of how interneurons drive oscillations must account for this diversity. This necessitates an investigation of how changing interneuron properties might disrupt the predictions of ING and PING, and whether other mechanisms might interact with or disrupt these network-driven mechanisms. In my dissertation, I broach this topic utilizing the Type I and Type II neuron classifications, which refer to properties derived from the mathematics of coupled oscillators. Classic ING and PING literature typically utilize Type I neurons which always respond to an excitatory perturbation with an advance of the subsequent action potential. However, many interneurons exhibit Type II properties, which respond to some excitatory perturbations with a delay in the subsequent action potential. Interneuronal diversity is also reflected in the strength and density of the synaptic connections between these neurons, which is also explored in this work. My research reveals a variety of ways in which interneuronal diversity alters synchronous oscillations in networks containing inhibitory interneurons and the mechanisms likely driving these dynamics. For example, oscillations in networks of Type II interneurons violate ING predictions and can be explained mechanistically primarily utilizing cellular properties. Additionally, varying the type of both excitatory and inhibitory cells in E-I networks reveals that synchronous excitatory activity arises with different network connectivities for different neuron types, sometimes driven by cellular properties rather than PING. Furthermore, E-I networks respond differently to varied strengths of inhibitory intra-connectivity depending upon interneuron type, sometimes in ways not fully accounted for by PING theory. Taken together, this research reveals that network-driven and cellularly-driven mechanisms promoting oscillatory activity in networks containing inhibitory interneurons interact, and oftentimes compete, in order to dictate the overall network dynamics. These dynamics are more complex than those predicted by the classic ING and PING mechanisms alone. The diverse dynamical properties imparted to oscillating neural networks by changing inhibitory interneuron properties provides some insight into the biological need for such variability.PHDApplied and Interdisciplinary MathematicsUniversity of Michigan, Horace H. Rackham School of Graduate Studieshttps://deepblue.lib.umich.edu/bitstream/2027.42/143981/1/sbrich_1.pd