Subthreshold dynamics of a single neuron from a Hamiltonian perspective

We use Hamilton's equations of classical mechanics to investigate the behavior of a cortical neuron on the approach to an action potential. We use a two-component dynamic model of a single neuron, due to Wilson, with added noise inputs. We derive a Lagrangian for the system, from which we construct Hamilton's equations. The conjugate momenta are found to be linear combinations of the noise input to the system. We use this approach to consider theoretically and computationally the most likely manner in which such a modeled neuron approaches a firing event. We find that the firing of a neuron is a result of a drop in inhibition, due to a temporary increase in negative bias of the mean noise input to the inhibitory control equation. Moreover, we demonstrate through theory and simulation that, on average, the bias in the noise increases in an exponential manner on the approach to an action potential. In the Hamiltonian description, an action potential can therefore be considered a result of the exponential growth of the conjugate momenta variables pulling the system away from its equilibrium state, into a nonlinear regime

Phase transitions in single neurons and neural populations: Critical slowing, anesthesia, and sleep cycles

The firing of an action potential by a biological neuron represents a dramatic transition from small-scale linear stochastics (subthreshold voltage fluctuations) to gross-scale nonlinear dynamics (birth of a 1-ms voltage spike). In populations of neurons we see similar, but slower, switch-like there-and-back transitions between low-firing background states and high-firing activated states. These state transitions are controlled by varying levels of input current (single neuron), varying amounts of GABAergic drug (anesthesia), or varying concentrations of neuromodulators and neurotransmitters (natural sleep), and all occur within a milieu of unrelenting biological noise. By tracking the altering responsiveness of the excitable membrane to noisy stimulus, we can infer how close the neuronal system (single unit or entire population) is to switching threshold. We can quantify this “nearness to switching” in terms of the altering eigenvalue structure: the dominant eigenvalue approaches zero, leading to a growth in correlated, low-frequency power, with exaggerated responsiveness to small perturbations, the responses becoming larger and slower as the neural population approaches its critical point–-this is critical slowing. In this chapter we discuss phase-transition predictions for both single-neuron and neural-population models, comparing theory with laboratory and clinical measurement

Characteristics of temporal fluctuations in the hyperpolarized state of the cortical slow oscillation

We present evidence for the hypothesis that transitions between the low- and high-firing states of the cortical slow oscillation correspond to neuronal phase transitions. By analyzing intracellular recordings of the membrane potential during the cortical slow oscillation in rats, we quantify the temporal fluctuations in power and the frequency centroid of the power spectrum in the period of time before “down” to “up” transitions. By taking appropriate averages over such events, we present these statistics as a function of time before transition. The results demonstrate an increase in fluctuation power and time scale broadly consistent with the slowing of systems close to phase transitions. The analysis is complicated and limited by the difficulty in identifying when transitions begin, and removing dc trends in membrane potential

Cortical patterns and gamma genesis are modulated by reversal potentials and gap-junction diffusion

In this chapter we describe a continuum model for the cortex that includes both axon-to-dendrite chemical synapses and direct neuron-to-neuron gap-junction diffusive synapses. The effectiveness of chemical synapses is determined by the voltage of the receiving dendrite V relative to its Nernst reversal potential Vrev. Here we explore two alternative strategies for incorporating dendritic reversal potentials, and uncover surprising differences in their stability properties and model dynamics. In the “slow-soma” variant, the (Vrev - V) weighting is applied after the input flux has been integrated at the dendrite, while for “fast-soma”, the weighting is applied directly to the input flux, prior to dendritic integration. For the slow-soma case, we find that–-provided the inhibitory diffusion (via gap-junctions) is sufficiently strong–-the cortex generates stationary Turing patterns of cortical activity. In contrast, the fast-soma destabilizes in favor of standing-wave spatial structures that oscillate at low-gamma frequency ( 30-Hz); these spatial patterns broaden and weaken as diffusive coupling increases, and disappear altogether at moderate levels of diffusion. We speculate that the slow- and fast-soma models might correspond respectively to the idling and active modes of the cortex, with slow-soma patterns providing the default background state, and emergence of gamma oscillations in the fast-soma case signaling the transition into the cognitive state

Measuring the electrical impedance of mouse brain tissue

We report on an experimental method to measure conductivity of cortical tissue. We use a pair of 5mm diameter Ag/AgCl electrodes in a Perspex sandwich device that can be brought to a distance of 400 microns apart. The apparatus is brought to uniform temperature before use. Electrical impedance of a sample is measured across the frequency range 20 Hz-2.0 MHz with an Agilent 4980A four-point impedance monitor in a shielded room. The equipment has been used to measure the conductivity of mature mouse brain cortex in vitro. Slices 400 microns in thickness are prepared on a vibratome. Slices are bathed in artificial cerebrospinal fluid (ACSF) to keep them alive. Slices are removed from the ACSF and sections of cortical tissue approximately 2 mm times 2 mm are cut with a razor blade. The sections are photographed through a calibrated microscope to allow identification of their cross-sectional areas. Excess ACSF is removed from the sample and the sections places between the electrodes. The impedance is measured across the frequency range and electrical conductivity calculated. Results show two regions of dispersion. A low frequency region is evident below approximately 10 kHz, and a high frequency dispersion above this. Results at the higher frequencies show a good fit to the Cole-Cole model of impedance of biological tissue; this model consists of resistive and non-linear capacitive elements. Physically, these elements are likely to arise due to membrane polarization and migration of ions both intra- and extra-cellularly.http://www.iupab2014.org/assets/IUPAB/NewFolder/iupab-abstracts.pd

Instabilities of the cortex during natural sleep

The electrical signals generated by the human cortex during sleep have been widely studied over the last 50 years. The electroencephalogram (EEG) observed during natural sleep exhibits structures with frequencies from 0.5 Hz to over 50 Hz and complicated waveforms such as spindles and K-complexes. Understanding has been enhanced by comprehensive intra-cellular measurements from the cortex and thalamus such as those performed by Steriade et al [1] and Sanchez-Vives and McCormick [2]. Models of the cerebal cortex have been developed in order to explain many of the features observed. These can be classified in terms of individual neuron models or collective models. Since we wish to compare predictions with gross features of the human EEG, we choose a collective model, where we average over a population of neurons in macrocolumns. A number of models of this form have been developed recently; that developed at Waikato draws from a number of different sources to describe the temporal and spatial dynamics of the system

A continuum model for the dynamics of the phase transition from slow-wave sleep to REM sleep

Previous studies have shown that activated cortical states (awake and rapid eye-movement (REM) sleep), are associated with increased cholinergic input into the cerebral cortex. However, the mechanisms that underlie the detailed dynamics of the cortical transition from slow-wave to REM sleep have not been quantitatively modeled. How does the sequence of abrupt changes in the cortical dynamics (as detected in the electrocorticogram) result from the more gradual change in subcortical cholinergic input? We compare the output from a continuum model of cortical neuronal dynamics with experimentally-derived rat electrocorticogram data. The output from the computer model was consistent with experimental observations. In slow-wave sleep, 0.5–2-Hz oscillations arise from the cortex jumping between “up” and “down” states on the stationary-state manifold. As cholinergic input increases, the upper state undergoes a bifurcation to an 8-Hz oscillation. The coexistence of both oscillations is similar to that found in the intermediate stage of sleep of the rat. Further cholinergic input moves the trajectory to a point where the lower part of the manifold in not available, and thus the slow oscillation abruptly ceases (REM sleep). The model provides a natural basis to explain neuromodulator-induced changes in cortical activity, and indicates that a cortical phase change, rather than a brainstem “flip-flop”, may describe the transition from slow-wave sleep to REM

What can a mean-field model tell us about the dynamics of the cortex?

In this chapter we examine the dynamical behavior of a spatially homogeneous two-dimensional model of the cortex that incorporates membrane potential, synaptic flux rates and long- and short-range synaptic input, in two spatial dimensions, using parameter sets broadly realistic of humans and rats. When synaptic dynamics are included, the steady states may not be stable. The bifurcation structure for the spatially symmetric case is explored, identifying the positions of saddle–node and sub- and supercritical Hopf instabilities. We go beyond consideration of small-amplitude perturbations to look at nonlinear dynamics. Spatially-symmetric (breathing mode) limit cycles are described, as well as the response to spatially-localized impulses. When close to Hopf and saddle–node bifurcations, such impulses can cause traveling waves with similarities to the slow oscillation of slow-wave sleep. Spiral waves can also be induced. We compare model dynamics with the known behavior of the cortex during natural and anesthetic-induced sleep, commenting on the physiological significance of the limit cycles and impulse responses

On the spatial Markov property of soups of unoriented and oriented loops

We describe simple properties of some soups of unoriented Markov loops and of some soups of oriented Markov loops that can be interpreted as a spatial Markov property of these loop-soups. This property of the latter soup is related to well-known features of the uniform spanning trees (such as Wilson's algorithm) while the Markov property of the former soup is related to the Gaussian Free Field and to identities used in the foundational papers of Symanzik, Nelson, and of Brydges, Fr\"ohlich and Spencer or Dynkin, or more recently by Le Jan

A kinetic and theoretical study of the borate catalysed reactions of hydrogen peroxide: the role of dioxaborirane as the catalytic intermediate for a wide range of substrates

Our recent work has provided new insights into the equilibria and species that exist in aqueous solution at different pHs for the boric acid – hydrogen peroxide system, and the role of these species in oxidation reactions. Most recently, (M. C. Durrant, D. M. Davies and M. E. Deary, Org. Biomol. Chem., 2011, 9,7249–7254), we have produced strong theoretical and experimental evidence for the existence of a previously unreported monocyclic three membered peroxide species, dioxaborirane, that is the likely catalytic species in borate mediated electrophilic reactions of hydrogen peroxide in alkaline solution. In the present paper, we extend our study of the borate–peroxide system to look at a wide range of substrates that include substituted dimethyl anilines, methyl-p-tolyl sulfoxide, halides, hydrogen sulfide anion, thiosulfate ,thiocyanate, and hydrazine. The unusual selectivity–reactivity pattern of borate catalysed reactions compared with hydrogen peroxide and inorganic or organic peracids previously observed for theorganic sulfides (D. M. Davies, M. E. Deary, K. Quill and R. A. Smith, Chem.–Eur. J., 2005, 11, 3552–3558) is also seen with substituted dimethyl aniline nucleophiles. This provides evidence that the pattern is not due to any latent electrophilic tendency of the organic sulfides and further supports dioxaborirane being the likely reactive intermediate, thus broadening the applicability of this catalytic system. Moreover, density functional theory calculations on our proposed mechanism involving dioxaborirane are consistent with the experimental results for these substrates. Results obtained at high concentrations of both borate and hydrogen peroxide require the inclusion the diperoxodiborate dianion in the kinetic analysis .A scheme detailing our current understanding of the borate–peroxide system is presented