Firing-Rate Models in Computational Neuroscience: New Applications and Methodologies

Abstract

Neurons make the nervous system tick. By communicating via electrical impulses, neurons form concentrated networks in localized regions of the brain to carry out the complex tasks required of the nervous system, ranging from moving the limbs to coordinating circadian rhythms. Despite their complexity, these networks encode much information in their rates of electrical impulse production–their firing rates. In Chapter 1, we review firing-rate models. We highlight several models that spearheaded the development of the field, as well as certain applications. We augment our review with an analysis of the corresponding network of citations. In doing so, we quantify how key papers contribute to the literature. We further classify firing-rate models according to modeling methodology. In Chapter 2, we provide a new sensitivity analysis methodology which we use to study a common type of chronic pain, allodynia, where non-painful stimuli produce pain. To do so, we employ coupled firing-rate models to understand two biophysically motivated circuit structures that represent common motifs within the dorsal horn of the spinal cord. The circuit motifs, respectively, regulate the production of static and dynamic allodynia, wherein gentle pressure (static) and gentle brushing sensations (dynamic) cause pain. To investigate variability in allodynia in each circuit motif, we identify the sets of coupling strengths that produce experimentally observed behaviors. To identify how properly behaving circuits are most vulnerable towards producing allodynia, we compute the minimal alteration in coupling strengths needed to induce the circuits to produce allodynia. We cluster the properly behaving circuits accordingly. Results indicate that in each circuit motif, allodynia is caused by unbalancing excitation and inhibition. Results further clarify how differences in coupling strengths or circuit structure lead to different vulnerabilities towards producing allodynia. In Chapter 3, we introduce a new firing-rate model formalism capable of simultaneously addressing multiple sources of heterogeneity in a neuronal network. In particular, we apply our model to the suprachiasmatic nucleus (SCN), which likely coordinates clocks throughout the body via 24hr oscillations in its firing rates. Further, SCN neurons intrinsically exhibit heterogeneous properties and various and non-standard forms of electrical impulses. Our formalism consists of a system of integro-differential equations describing the time evolution of the mean and standard deviation of synaptic conductances across the network. Properties of SCN neurons are incorporated by computing responses to synaptic conductance inputs of a Hodgkin-Huxley-type SCN neuron model that exhibits these non-standard firing patterns. Such responses are then averaged over distributions of relevant quantities and included in the differential equations. Results suggest mechanisms by which physiologically relevant changes to firing rates may appear. For instance, results show that a large spread in circadian phases across SCN neurons reduces the amplitude of the 24hr oscillations in SCN network firing activity, identifying a mechanism by which heterogeneities in neuron electrophysiology could influence circadian rhythms. In Chapter 4, we employ the firing-rate model from Chapter 3 to mechanistically understand how environmental light, e.g. sunlight and smartphone light, affects SCN firing activity. In doing so, we find that we could better describe SCN output than if we use standard firing-rate models. Further, in modeling the light-to-SCN-output pathway, we identify a novel trajectory traversed throughout the day by conductances of important potassium ion channels in SCN neurons. The resulting trajectory clarifies the link between molecular clocks within SCN neurons to the electrical state of the neurons.