This article reviews the electroresponsive properties of single neurons in the mammalian central nervous system (CNS). In some of these cells the ionic conductances responsible for their excitability also endow them with autorhythmic electrical oscillatory properties. Chemical or electrical synaptic contacts between these neurons often result in network oscillations. In such networks, autorhythmic neurons may act as true oscillators (as pacemakers) or as resonators (responding preferentially to certain firing frequencies). Oscillations and resonance in the CNS are proposed to have diverse functional roles, such as (i) determining global functional states (for example, sleep-wakefulness or attention), (ii) timing in motor coordination, and (iii) specifying connectivity during development. Also, oscillation, especially in the thalamo-cortical circuits, may be related to certain neurological and psychiatric disorders. This review proposes that the autorhythmic electrical properties of central neurons and their connectivity form the basis for an intrinsic functional coordinate system that provides internal context to sensory input.

https://science.sciencemag.org/content/sci/242/4886/1654.full.pdf

The intrinsic electrophysiological properties of mammalian neurons: insights into central nervous system function.

Bifurcation mechanisms involved in the generation of action potentials (spikes) by neurons are reviewed here. We show how the type of bifurcation determines the neuro-computational properties of the cells. For example, when the rest state is near a saddle-node bifurcation, the cell can fire all-or-none spikes with an arbitrary low frequency, it has a well-defined threshold manifold, and it acts as an integrator; i.e. the higher the frequency of incoming pulses, the sooner it fires. In contrast, when the rest state is near an Andronov–Hopf bifurcation, the cell fires in a certain frequency range, its spikes are not all-or-none, it does not have a well-defined threshold manifold, it can fire in response to an inhibitory pulse, and it acts as a resonator; i.e. it responds preferentially to a certain (resonant) frequency of the input. Increasing the input frequency may actually delay or terminate its firing. We also describe the phenomenon of neural bursting, and we use geometric bifurcation theory to extend the existing classification of bursters, including many new types. We discuss how the type of burster defines its neuro-computational properties, and we show that different bursters can interact, synchronize and process information differently.

/pdf/neural-excitability-spiking-and-bursting-2ivtoscvw8.pdf

Neural excitability, spiking and bursting

http://brainimaging.waisman.wisc.edu/~lutz/LeVanQuyen_et_all_JNM_2001.pdf

Comparison of Hilbert transform and wavelet methods for the analysis of neuronal synchrony

Cheetahs and beetles run, dolphins and salmon swim, and bees and birds fly with grace and economy surpassing our technology. Evolution has shaped the breathtaking abilities of animals, leaving us the challenge of reconstructing their targets of control and mechanisms of dexterity. In this review we explore a corner of this fascinating world. We describe mathematical models for legged animal locomotion, focusing on rapidly running insects and highlighting past achievements and challenges that remain. Newtonian body--limb dynamics are most naturally formulated as piecewise-holonomic rigid body mechanical systems, whose constraints change as legs touch down or lift off. Central pattern generators and proprioceptive sensing require models of spiking neurons and simplified phase oscillator descriptions of ensembles of them. A full neuromechanical model of a running animal requires integration of these elements, along with proprioceptive feedback and models of goal-oriented sensing, planning, and learning. We outline relevant background material from biomechanics and neurobiology, explain key properties of the hybrid dynamical systems that underlie legged locomotion models, and provide numerous examples of such models, from the simplest, completely soluble "peg-leg walker" to complex neuromuscular subsystems that are yet to be assembled into models of behaving animals. This final integration in a tractable and illuminating model is an outstanding challenge.

/pdf/the-dynamics-of-legged-locomotion-models-analyses-and-4e8knboeub.pdf

The Dynamics of Legged Locomotion: Models, Analyses, and Challenges

This work was supported by NSF Grant No. NSF/EIA-0130708, and Grant No. PHY 0414174; NIH Grant No. 1 R01 NS50945 and Grant No. NS40110; MEC BFI2003-07276, and Fundacion BBVA.

/pdf/dynamical-principles-in-neuroscience-3b8nfk3ipm.pdf

Dynamical principles in neuroscience

We describe a modification to our recent model of the action potential which introduces two additional equilibrium points. By using stability analysis we show that one of these equilibrium points is a saddle point from which there are two separatrices which divide the phase plane into two regions. In one region all phase paths approach a limit cycle and in the other all phase paths approach a stable equilibrium point. A consequence of this is that a short depolarizing current pulse will change an initially silent model neuron into one that fires repetitively. Addition of a third equation limits this firing to either an isolated burst or a depolarizing afterpotential. When steady depolarizing current was applied to this model it resulted in periodic bursting. The equations, which were initially developed to explain isolated triggered bursts, therefore provide one of the simplest models of the more general phenomenon of oscillatory burst discharge.

/pdf/a-model-of-neuronal-bursting-using-three-coupled-first-order-56eke0sz43.pdf

A model of neuronal bursting using three coupled first order differential equations.

We modify our recent three equilibrium-point model of neuronal bursting by a means of a small deformation of the nullclines in the x-y phase plane to give a model that can have as many as five equilibrium points. In this model the middle stable equilibrium point (e.p.) is separated from the outer stable and unstable e.ps by two saddle points. If the system is started at rest at the middle stable e.p. it has the following complex properties: A short suprathreshold current pulse switches the model from a silent state to a bursting state, or to give a single burst, depending on the choice of parameters. A subthreshold depolarizing current step gives a passive response at rest, but if the model is either constantly hyperpolarized or constantly depolarized, then the same current step gives different active responses. At a hyperpolarized level this consists of a burst response that shows refractoriness. At a depolarized level it consists of tonic firing with a linear frequency--current relationship. Hyperpolarization from rest is followed by post-inhibitory rebound. The model responds in a unique and characteristic way to an applied current ramp. These properties are very similar to those that have been recently recorded intracellularly from neurons in the mammalian thalamus. In the x-y phase plane our models of the repetitively firing neuron, the bursting neuron and the thalamic neuron form a progression of models in which the y nullcline in the subthreshold region is deformed once to give the burst neuron model, and a second time to give the thalamic neuron model. Each deformation can be interpreted as corresponding to the inclusion of a slow inward current in the model. As these currents are included so the associated firing properties increase in complexity.

A model of a thalamic neuron.

In this present work polynomial transformations are identified that preserve the property of the polynomials having all zeros lying on the imaginary axis. Existence results concerning families of polynomials whose generalized Mellin transforms have zeros all lying on the critical line are then derived. Inherent structures are identified from which a simple proof relating to the Gegenbauer family of orthogonal polynomials is subsequently deduced. Some discussion about the choice of generalized Mellin transform is also given.

J. L. Hindmarsh

Papers

A model of neuronal bursting using three coupled first order differential equations.

A model of a thalamic neuron.

On polynomial transformations preserving purely imaginary zeros*