John R. Cressman

The influence of sodium and potassium dynamics on excitability, seizures, and the stability of persistent states: I. Single neuron dynamics

In these companion papers, we study how the interrelated dynamics of sodium and potassium affect the excitability of neurons, the occurrence of seizures, and the stability of persistent states of activity. In this first paper, we construct a mathematical model consisting of a single conductance-based neuron together with intra- and extracellular ion concentration dynamics. We formulate a reduction of this model that permits a detailed bifurcation analysis, and show that the reduced model is a reasonable approximation of the full model. We find that competition between intrinsic neuronal currents, sodium-potassium pumps, glia, and diffusion can produce very slow and large-amplitude oscillations in ion concentrations similar to what is seen physiologically in seizures. Using the reduced model, we identify the dynamical mechanisms that give rise to these phenomena. These models reveal several experimentally testable predictions. Our work emphasizes the critical role of ion concentration homeostasis in the proper functioning of neurons, and points to important fundamental processes that may underlie pathological states such as epilepsy.

The influence of sodium and potassium dynamics on excitability, seizures, and the stability of persistent states: II. Network and glial dynamics

In these companion papers, we study how the interrelated dynamics of sodium and potassium affect the excitability of neurons, the occurrence of seizures, and the stability of persistent states of activity. We seek to study these dynamics with respect to the following compartments: neurons, glia, and extracellular space. We are particularly interested in the slower time-scale dynamics that determine overall excitability, and set the stage for transient episodes of persistent oscillations, working memory, or seizures. In this second of two companion papers, we present an ionic current network model composed of populations of Hodgkin–Huxley type excitatory and inhibitory neurons embedded within extracellular space and glia, in order to investigate the role of micro-environmental ionic dynamics on the stability of persistent activity. We show that these networks reproduce seizure-like activity if glial cells fail to maintain the proper micro-environmental conditions surrounding neurons, and produce several experimentally testable predictions. Our work suggests that the stability of persistent states to perturbation is set by glial activity, and that how the response to such perturbations decays or grows may be a critical factor in a variety of disparate transient phenomena such as working memory, burst firing in neonatal brain or spinal cord, up states, seizures, and cortical oscillations.

Ion concentration dynamics as a mechanism for neuronal bursting

We describe a simple conductance-based model neuron that includes intra- and extracellular ion concentration dynamics and show that this model exhibits periodic bursting. The bursting arises as the fast-spiking behavior of the neuron is modulated by the slow oscillatory behavior in the ion concentration variables and vice versa. By separating these time scales and studying the bifurcation structure of the neuron, we catalog several qualitatively different bursting profiles that are strikingly similar to those seen in experimental preparations. Our work suggests that ion concentration dynamics may play an important role in modulating neuronal excitability in real biological systems.

Eulerian and Lagrangian studies in surface flow turbulence

Experimental and numerical studies of turbulent fluid motion in a free surface are presented. The flow is realized experimentally on the surface of a tank filled with water stirred well below the surface. Numerically, it is modelled by free-slip boundary conditions. The surface flow is unconventional: it is not incompressible and neither kinetic energy nor enstrophy is conserved in the limit of zero fluid viscosity and in the absence of external driving as is the case for incompressible two-dimensional turbulent flows. The dynamics of passive Lagrangian tracers that are advected in such flows are dominated by rapidly changing patches of the surface flow divergence. Owing to compressibility, particles form clusters within multifractal mass distributions. Also studied is the motion of pairs and triplets of particles. The mean square separation shows an extended range with a reduced scaling exponent in comparison with the classical Richardson value. Clustering is also manifest in strongly deformed triangles spanned within triplets of tracers.

Seizures as imbalanced up states: excitatory and inhibitory conductances during seizure-like events

Precisely timed and dynamically balanced excitatory (E) and inhibitory (I) conductances underlie the basis of neural network activity. Normal E/I balance is often shifted in epilepsy, resulting in neuronal network hyperexcitability and recurrent seizures. However, dynamics of the actual excitatory and inhibitory synaptic conductances (ge and gi, respectively) during seizures remain unknown. To study the dynamics of E and I network balance, we calculated ge and gi during the initiation, body, and termination of seizure-like events (SLEs) in the rat hippocampus in vitro. Repetitive emergent SLEs in 4-aminopyridine (100 μM) and reduced extracellular magnesium (0.6 mM) were recorded in the identified CA1 pyramidal cells (PC) and oriens-lacunosum moleculare (O-LM) interneurons. Calculated ge/gi ratio dynamics showed that the initiation stage of the SLEs was dominated by inhibition in the PCs and was more balanced in the O-LM cells. During the body of the SLEs, the balance shifted toward excitation, with ge and gi peaking in both cell types at nearly the same time. In the termination phase, PCs were again dominated by inhibition, whereas O-LM cells experienced persistent excitatory synaptic barrage. In this way, increased excitability of interneurons may play roles in both seizure initiation (žiburkus J, Cressman JR, Barreto E, Schiff SJ. J Neurophysiol 95: 3948-3954, 2006) and in their termination. Overall, SLE stages can be characterized in PC and O-LM cells by dynamically distinct changes in the balance of ge and gi, where a temporal sequence of imbalance shifts with the changing firing patterns of the cellular subtypes comprising the hyperexcitable microcircuits.

Time-Scale Separation from Diffusion-Mapped Delay Coordinates

It has long been known that the method of time-delay embedding can be used to reconstruct nonlinear dynamics from time series data. A less-appreciated fact is that the induced geometry of time-delay coordinates increasingly biases the reconstruction toward the stable directions as delays are added. This bias can be exploited, using the diffusion maps approach to dimension reduction, to extract dynamics on desired time scales from high-dimensional observed data. We demonstrate the technique on a wide range of examples, including data generated by a model of meandering spiral waves and video recordings of a liquid-crystal experiment.

Oxygen and seizure dynamics: I. Experiments

We utilized a novel ratiometric nanoquantum dot fluorescence resonance energy transfer (NQD-FRET) optical sensor to quantitatively measure oxygen dynamics from single cell microdomains during hypoxic episodes as well as during 4-aminopyridine (4-AP)-induced spontaneous seizure-like events in rat hippocampal slices. Coupling oxygen sensing with electrical recordings, we found the greatest reduction in the O2 concentration ([O2]) in the densely packed cell body stratum (st.) pyramidale layer of the CA1 and differential layer-specific O2 dynamics between the st. pyramidale and st. oriens layers. These hypoxic decrements occurred up to several seconds before seizure onset could be electrically measured extracellularly. Without 4-AP, we quantified a narrow range of [O2], similar to the endogenous hypoxia found before epileptiform activity, which permits a quiescent network to enter into a seizure-like state. We demonstrated layer-specific patterns of O2 utilization accompanying layer-specific neuronal interplay in seizure. None of the oxygen overshoot artifacts seen with polarographic measurement techniques were observed. We therefore conclude that endogenously generated hypoxia may be more than just a consequence of increased cellular excitability but an influential and critical factor for orchestrating network dynamics associated with epileptiform activity.

Modification of a vortex street by a polymer additive

A Karman vortex street is created in a flowing soap film by a rod penetrating the film. The velocity field generated by this rod is modified by the addition of the polymer polyethylene oxide having a molecular weight of 5×106 and a concentration of 30 wppm. The rms velocity fluctuations behind the rod are strongly suppressed by the polymer additive and the power spectrum of the velocity fluctuations is modified as well. The experiments show that the polymer additive decreases the rate at which energy is injected into the flow. The measurements further indicate that the polymer introduces an elongational viscosity term into the Navier–Stokes equation.

Desynchronization of Fast-Spiking Interneurons Reduces β-Band Oscillations and Imbalance in Firing in the Dopamine-Depleted Striatum

Oscillations in the β-band (8–30 Hz) that emerge in the output nuclei of the basal ganglia during Parkinsons disease, along with an imbalanced activation of the direct and indirect pathways, have been linked to the hypokinetic motor output associated with the disease. Although dopamine depletion causes a change in cellular and network properties in the striatum, it is unclear whether abnormal activity measured in the globus pallidus and substantia nigra pars reticulata is caused by abnormal striatal activity. Here we use a computational network model of medium spiny neurons (MSNs)—fast-spiking interneurons (FSIs), based on data from several mammalian species, and find that robust β-band oscillations and imbalanced firing emerge from implementation of changes to cellular and circuit properties caused by dopamine depletion. These changes include a reduction in connections between MSNs, a doubling of FSI inhibition to D2 MSNs, an increase in D2 MSN dendritic excitability, and a reduction in D2 MSN somatic excitability. The model reveals that the reduced decorrelation between MSNs attributable to weakened lateral inhibition enables the strong influence of synchronous FSIs on MSN firing and oscillations. Weakened lateral inhibition also produces an increased sensitivity of MSN output to cortical correlation, a condition relevant to the parkinsonian striatum. The oscillations of FSIs, in turn, are strongly modulated by fast electrical transmission between FSIs through gap junctions. These results suggest that pharmaceuticals that desynchronize FSI activity may provide a novel treatment for the enhanced β-band oscillations, imbalanced firing, and motor dysfunction in Parkinsons disease.

Erratum to: The influence of sodium and potassium dynamics on excitability, seizures, and the stability of persistent states: I. Single neuron dynamics

The original version of this article contains a few typographical errors. These do not affect the results and conclusions that we originally reported. However, to assist readers interested in reproducing our results, we list below several corrections in order to accurately reflect the equations that were used to generate the published figures. Some errors are consequential, and others are less so; we list them all for completeness. Code containing the correct equations is available at ModelDB (http://senselab.med.yale.edu/modeldb). Equation (2) should appear as follows: