Rhythms of the nervous system: mathematical themes and variations

The nervous system displays a variety of rhythms in both waking and sleep. These rhythms have been closely associated with different behavioral and cognitive states, but it is still unknown how the nervous system makes use of these rhythms to perform functionally important tasks. To address those questions, it is first useful to understood in a mechanistic way the origin of the rhythms, their interactions, the signals which create the transitions among rhythms, and the ways in which rhythms filter the signals to a network of neurons. This talk discusses how dynamical systems have been used to investigate the origin, properties and interactions of rhythms in the nervous system. It focuses on how the underlying physiology of the cells and synapses of the networks shape the dynamics of the network in different contexts, allowing the variety of dynamical behaviors to be displayed by the same network. The work is presented using a series of related case studies on different rhythms. These case studies are chosen to highlight mathematical issues, and suggest further mathematical work to be done. The topics include: different roles of excitation and inhibition in creating synchronous assemblies of cells, different kinds of building blocks for neural oscillations, and transitions among rhythms. The mathematical issues include reduction of large networks to low dimensional maps, role of noise, global bifurcations, use of probabilistic formulations.Published versio

Approximate, not perfect synchrony maximizes the downstream effectiveness of excitatory neuronal ensembles

The most basic functional role commonly ascribed to synchrony in the brain is that of amplifying excitatory neuronal signals. The reasoning is straightforward: When positive charge is injected into a leaky target neuron over a time window of positive duration, some of it will have time to leak back out before an action potential is triggered in the target, and it will in that sense be wasted. If the goal is to elicit a firing response in the target using as little charge as possible, it seems best to deliver the charge all at once, i.e., in perfect synchrony. In this article, we show that this reasoning is correct only if one assumes that the input ceases when the target crosses the firing threshold, but before it actually fires. If the input ceases later-for instance, in response to a feedback signal triggered by the firing of the target-the "most economical" way of delivering input (the way that requires the least total amount of input) is no longer precisely synchronous, but merely approximately so. If the target is a heterogeneous network, as it always is in the brain, then ceasing the input "when the target crosses the firing threshold" is not an option, because there is no single moment when the firing threshold is crossed. In this sense, precise synchrony is never optimal in the brain.R01 NS067199 - NINDS NIH HH

Excitable neurons, firing threshold manifolds and canards

We investigate firing threshold manifolds in a mathematical model of an excitable neuron. The model analyzed investigates the phenomenon of post-inhibitory rebound spiking due to propofol anesthesia and is adapted from McCarthy et al. (SIAM J. Appl. Dyn. Syst. 11(4):1674-1697, 2012). Propofol modulates the decay time-scale of an inhibitory GABAa synaptic current. Interestingly, this system gives rise to rebound spiking within a specific range of propofol doses. Using techniques from geometric singular perturbation theory, we identify geometric structures, known as canards of folded saddle-type, which form the firing threshold manifolds. We find that the position and orientation of the canard separatrix is propofol dependent. Thus, the speeds of relevant slow synaptic processes are encoded within this geometric structure. We show that this behavior cannot be understood using a static, inhibitory current step protocol, which can provide a single threshold for rebound spiking but cannot explain the observed cessation of spiking for higher propofol doses. We then compare the analyses of dynamic and static synaptic inhibition, showing how the firing threshold manifolds of each relate, and why a current step approach is unable to fully capture the behavior of this model

Frequency control in synchronized networks of inhibitory neurons

We analyze the control of frequency for a synchronized inhibitory neuronal network. The analysis is done for a reduced membrane model with a biophysically-based synaptic influence. We argue that such a reduced model can quantitatively capture the frequency behavior of a larger class of neuronal models. We show that in different parameter regimes, the network frequency depends in different ways on the intrinsic and synaptic time constants. Only in one portion of the parameter space, called `phasic', is the network period proportional to the synaptic decay time. These results are discussed in connection with previous work of the authors, which showed that for mildly heterogeneous networks, the synchrony breaks down, but coherence is preserved much more for systems in the phasic regime than in the other regimes. These results imply that for mildly heterogeneous networks, the existence of a coherent rhythm implies a linear dependence of the network period on synaptic decay time, and a much weaker dependence on the drive to the cells. We give experimental evidence for this conclusion.Comment: 18 pages, 3 figures, Kluwer.sty. J. Comp. Neurosci. (in press). Originally submitted to the neuro-sys archive which was never publicly announced (was 9803001

Striatal cholinergic interneurons generate beta and gamma oscillations in the corticostriatal circuit and produce motor deficits

Cortico-basal ganglia-thalamic (CBT) neural circuits are critical modulators of cognitive and motor function. When compromised, these circuits contribute to neurological and psychiatric disorders, such as Parkinson's disease (PD). In PD, motor deficits correlate with the emergence of exaggerated beta frequency (15-30 Hz) oscillations throughout the CBT network. However, little is known about how specific cell types within individual CBT brain regions support the generation, propagation, and interaction of oscillatory dynamics throughout the CBT circuit or how specific oscillatory dynamics are related to motor function. Here, we investigated the role of striatal cholinergic interneurons (SChIs) in generating beta and gamma oscillations in cortical-striatal circuits and in influencing movement behavior. We found that selective stimulation of SChIs via optogenetics in normal mice robustly and reversibly amplified beta and gamma oscillations that are supported by distinct mechanisms within striatal-cortical circuits. Whereas beta oscillations are supported robustly in the striatum and all layers of primary motor cortex (M1) through a muscarinic-receptor mediated mechanism, gamma oscillations are largely restricted to the striatum and the deeper layers of M1. Finally, SChI activation led to parkinsonian-like motor deficits in otherwise normal mice. These results highlight the important role of striatal cholinergic interneurons in supporting oscillations in the CBT network that are closely related to movement and parkinsonian motor symptoms.DP2 NS082126 - NINDS NIH HHS; R01 NS081716 - NINDS NIH HHS; R21 NS078660 - NINDS NIH HHShttps://www.ncbi.nlm.nih.gov/pmc/articles/PMC4896681/Published versio

A mechanism for the formation of hippocampal neuronal firing patterns that represent what happens where

The association of specific events with the context in which they occur is a fundamental feature of episodic memory. However, the underlying network mechanisms generating what–where associations are poorly understood. Recently we reported that some hippocampal principal neurons develop representations of specific events occurring in particular locations (item-position cells). Here, we investigate the emergence of item-position selectivity as rats learn new associations for reward and find that before the animal’s performance rises above chance in the task, neurons that will later become itemposition cells have a strong selective bias toward one of two behavioral responses, which the animal will subsequently make to that stimulus. This response bias results in an asymmetry of neural activity on correct and error trials that could drive the emergence of particular item specificities based on a simple reward-driven synaptic plasticity mechanism

Neurosystems: brain rhythms and cognitive processing

Neuronal rhythms are ubiquitous features of brain dynamics, and are highly correlated with cognitive processing. However, the relationship between the physiological mechanisms producing these rhythms and the functions associated with the rhythms remains mysterious. This article investigates the contributions of rhythms to basic cognitive computations (such as filtering signals by coherence and/or frequency) and to major cognitive functions (such as attention and multi-modal coordination). We offer support to the premise that the physiology underlying brain rhythms plays an essential role in how these rhythms facilitate some cognitive operations.098352 - Wellcome Trust; 5R01NS067199 - NINDS NIH HH

Are Different Rhythms Good for Different Functions?

This essay discusses the relationship between the physiology of rhythms and potential functional roles. We focus on how the biophysics underlying different rhythms can give rise to different abilities of a network to form and manipulate cell assemblies. We also discuss how changes in the modulatory setting of the rhythms can change the flow of information through cortical circuits, again tying physiology to computation. We suggest that diverse rhythms, or variations of a rhythm, can support different components of a cognitive act, with multiple rhythms potentially playing multiple roles

Interactions of multiple rhythms in a biophysical network of neurons

Neural oscillations, including rhythms in the beta1 band (12–20 Hz), are important in various cognitive functions. Often neural networks receive rhythmic input at frequencies different from their natural frequency, but very little is known about how such input affects the network’s behavior. We use a simplified, yet biophysical, model of a beta1 rhythm that occurs in the parietal cortex, in order to study its response to oscillatory inputs. We demonstrate that a cell has the ability to respond at the same time to two periodic stimuli of unrelated frequencies, firing in phase with one, but with a mean firing rate equal to that of the other. We show that this is a very general phenomenon, independent of the model used. We next show numerically that the behavior of a different cell, which is modeled as a high-dimensional dynamical system, can be described in a surprisingly simple way, owing to a reset that occurs in the state space when the cell fires. The interaction of the two cells leads to novel combinations of properties for neural dynamics, such as mode-locking to an input without phase-locking to it.Published versio

GABAB receptor-mediated, layer-specific synaptic plasticity reorganizes gamma-frequency neocortical response to stimulation

Repeated presentations of sensory stimuli generate transient gamma-frequency (30-80 Hz) responses in neocortex that show plasticity in a task-dependent manner. Complex relationships between individual neuronal outputs and the mean, local field potential (population activity) accompany these changes, but little is known about the underlying mechanisms responsible. Here we show that transient stimulation of input layer 4 sufficient to generate gamma oscillations induced two different, lamina-specific plastic processes that correlated with lamina-specific changes in responses to further, repeated stimulation: Unit rates and recruitment showed overall enhancement in supragranular layers and suppression in infragranular layers associated with excitatory or inhibitory synaptic potentiation onto principal cells, respectively. Both synaptic processes were critically dependent on activation of GABAB receptors and, together, appeared to temporally segregate the cortical representation. These data suggest that adaptation to repetitive sensory input dramatically alters the spatiotemporal properties of the neocortical response in a manner that may both refine and minimize cortical output simultaneously