Mode locking in a periodically forced "ghostbursting" neuron model

We study a minimal integrate-and-fire based model of a "ghostbursting" neuron under periodic stimulation. These neurons are involved in sensory processing in weakly electric fish. There exist regions in parameter space in which the model neuron is mode-locked to the stimulation. We analyse this locked behavior and examine the bifurcations that define the boundaries of these regions. Due to the discontinuous nature of the flow, some of these bifurcations are nonsmooth. This exact analysis is in excellent agreement with numerical simulations, and can be used to understand the response of such a model neuron to biologically realistic input.

Neuronal networks with gap junctions: A study of piece-wise linear planar neuron models

The presence of gap junction coupling among neurons of the central nervous systems has been appreciated for some time now. In recent years there has been an upsurge of interest from the mathematical community in understanding the contribution of these direct electrical connections between cells to large-scale brain rhythms. Here we analyze a class of exactly soluble single neuron models, capable of producing realistic action potential shapes, that can be used as the basis for understanding dynamics at the network level. This work focuses on planar piece-wise linear models that can mimic the firing response of several different cell types. Under constant current injection the periodic response and phase response curve (PRC) is calculated in closed form. A simple formula for the stability of a periodic orbit is found using Floquet theory. From the calculated PRC and the periodic orbit a phase interaction function is constructed that allows the investigation of phase-locked network states using the theory of weakly coupled oscillators. For large networks with global gap junction connectivity we develop a theory of strong coupling instabilities of the homogeneous, synchronous and splay state. For a piece-wise linear caricature of the Morris-Lecar model, with oscillations arising from a homoclinic bifurcation, we show that large amplitude oscillations in the mean membrane potential are organized around such unstable orbits.

Mode locking in a spatially extended neuron model: active soma and compartmental tree

Understanding the mode-locked response of excitable systems to periodic forcing has important applications in neuroscience. For example it is known that spatially extended place cells in the hippocampus are driven by the theta rhythm to generate a code conveying information about spatial location. Thus it is important to explore the role of neuronal dendrites in generating the response to periodic current injection. In this paper we pursue this using a compartmental model, with linear dynamics for each compartment, coupled to an active soma model that generates action potentials. By working with the piece-wise linear McKean model for the soma we show how the response of the whole neuron model (soma and dendrites) can be written in closed form. We exploit this to construct a stroboscopic map describing the response of the spatially extended model to periodic forcing. A linear stability analysis of this map, together with a careful treatment of the non-differentiability of the soma model, allows us to construct the Arnol'd tongue structure for 1:q states (one action potential for q cycles of forcing). Importantly we show how the presence of quasi-active membrane in the dendrites can influence the shape of tongues. Direct numerical simulations confirm our theory and further indicate that resonant dendritic membrane can enlarge the windows in parameter space for chaotic behavior. These simulations also show that the spatially extended neuron model responds differently to global as opposed to point forcing. In the former case spatio-temporal patterns of activity within an Arnol'd tongue are standing waves, whilst in the latter they are traveling waves.

Clustering through post inhibitory rebound in synaptically coupled neurons

Post inhibitory rebound is a nonlinear phenomenon present in a variety of nerve cells. Following a period of hyper-polarization this effect allows a neuron to fire a spike or packet of spikes before returning to rest. It is an important mechanism underlying central pattern generation for heartbeat, swimming and other motor patterns in many neuronal systems. In this paper we consider how networks of neurons, which do not intrinsically oscillate, may make use of inhibitory synaptic connections to generate large scale coherent rhythms in the form of cluster states. We distinguish between two cases i) where the rebound mechanism is due to anode break excitation and ii) where rebound is due to a slow T-type calcium current. In the former case we use a geometric analysis of a McKean type model to obtain expressions for the number of clusters in terms of the speed and strength of synaptic coupling. Results are found to be in good qualitative agreement with numerical simulations of the more detailed Hodgkin-Huxley model. In the second case we consider a particular firing rate model of a neuron with a slow calcium current that admits to an exact analysis. Once again existence regions for cluster states are explicitly calculated. Both mechanisms are shown to prefer globally synchronous states for slow synapses as long as the strength of coupling is sufficiently large. With a decrease in the duration of synaptic inhibition both systems are found to break into clusters. A major difference between the two mechanisms for cluster generation is that anode break excitation can support clusters with several groups, whilst slow T-type calcium currents predominantly give rise to clusters of just two (anti-synchronous) populations.

Bumps, breathers, and waves in a neural network with spike frequency adaptation

In this Letter we introduce a continuum model of neural tissue that include the effects of so-called spike frequency adaptation (SFA). The basic model is an integral equation for synaptic activity that depends upon the non-local network connectivity, synaptic response, and firing rate of a single neuron. A phenomenological model of SFA is examined whereby the firing rate is taken to be a simple state-dependent threshold function. As in the case without SFA classical Mexican-Hat connectivity is shown to allow for the existence of spatially localized states (bumps). Importantly an analysis of bump stability using recent Evans function techniques shows that bumps may undergo instabilities leading to the emergence of both breathers and traveling waves. Moreover, a similar analysis for traveling pulses leads to the conditions necessary to observe a stable traveling breather. Direct numerical simulations both confirm our theoretical predictions and illustrate the rich dynamic behavior of this model, including the appearance of self-replicating bumps.

Toward blueprints for network architecture, biophysical dynamics, and signal transduction

We review mathematical aspects of biophysical dynamics, signal transduction and network architecture that have been used to uncover functionally significant relations between the dynamics of single neurons and the networks they compose. We focus on examples that combine insights from these three areas to expand our understanding of systems neuroscience. These range from single neuron coding to models of decision making and electrosensory discrimination by networks and populations, as well as coincidence detection in pairs of dendrites and the dynamics of large networks of excitable dendritic spines. We conclude by describing some of the challenges that lie ahead as the applied mathematics community seeks to provide the tools that will ultimately underpin systems neuroscience.

Branching dendrites with resonant membrane: a "sum-over-trips" approach

Dendrites form the major components of neurons. They are complex branching structures that receive and process thousands of synaptic inputs from other neurons. It is well known that dendritic morphology plays an important role in the function of dendrites. Another important contribution to the response characteristics of a single neuron comes from the intrinsic resonant properties of dendritic membrane. In this paper we combine the effects of dendritic branching and resonant membrane dynamics by generalising the "sum-over-trips" approach [Abbott, L.F., Fahri, E., Gutmann, S.: The path integral for dendritic trees. Biological Cybernetics 66, 49--60 (1991)]. To illustrate how this formalism can shed light on the role of architecture and resonances in determining neuronal output we consider dual recording and reconstruction data from a rat CA1 hippocampal pyramidal cell. Specifically we explore the way in which an $I_{h}$ current contributes to a voltage overshoot at the soma.

Gap junctions and emergent rhythms

Gap junction coupling is ubiquitous in the brain, particularly between the dendritic trees of inhibitory interneurons. Such direct non-synaptic interaction allows for direct electrical communication between cells. Unlike spike-time driven synaptic neural network models, which are event based, any model with gap junctions must necessarily involve a single neuron model that can represent the shape of an action potential. Indeed, not only do neurons communicating via gaps feel super-threshold spikes, but they also experience, and respond to, sub-threshold voltage signals. In this chapter we show that the so-called absolute integrate-and-fire model is ideally suited to such studies. At the single neuron level voltage traces for the model may be obtained in closed form, and are shown to mimic those of fast-spiking inhibitory neurons. Interestingly in the presence of a slow spike adaptation current the model is shown to support periodic bursting oscillations. For both tonic and bursting modes the phase response curve can be calculated in closed form. At the network level we focus on global gap junction coupling and show how to analyze the asynchronous firing state in large networks. Importantly, we are able to determine the emergence of non-trivial network rhythms due to strong coupling instabilities. To illustrate the use of our theoretical techniques (particularly the phase-density formalism used to determine stability) we focus on a spike adaptation induced transition from asynchronous tonic activity to synchronous bursting in a gap-junction coupled network.