Variation of repetitive cortical spreading depression waves is related with relative refractory period: a computational study

Cortical spreading depression (CSD) is an important experimental model for diseases such as stroke, epilepsy and migraine. Previous observations indicated that the amplitude and velocity of the typical direct current potential shift during repetitive CSD waves were varying. The recovery state of the tissue was found related with the variation of successive CSD waves. A computational model in this paper aimed to investigate the role of relative refractory period of CSD. This model simulated that continuous injection of KCl solution induced repetitive CSD waves. The first CSD wave often had a larger amplitude and faster velocity than those of the succeeding secondary waves. The relative refractory period lasted much longer than the recovery of ions turbulence. If the induction interval was long enough for recovery, a series of CSD waves would have the same profile as the first one. In the relative refractory period, an early stimulation might lead to a late initiation of CSD, i.e., “haste makes waste”. The amplitude and velocity of CSD waves were found increasing with the initiation interval and asymptotic to those of the first CSD wave. This study verified that the propagation dynamics of CSD waves is modulated by the relative refractory period. It suggested that the refractory period is critical for preventing undesirable CSD waves.     

The mechanism of the initiation of cardiac arrhythmias due to a pathological distribution of myocardial conductivity

The two mechanisms of cardiac arrhythmia that are currently known involve transition of some of the cells into a self-oscillating mode and the emergence of circulating waves. A mechanism for the emergence of circulating waves due to a unidirectional blockage is considered in the present study. A narrow gap between two non-conductive areas may provide the conditions for such a mechanism. However, this mechanism cannot occur in the human heart, since the minimal path length for circulation at the action potential duration of 0.3 s and the wave propagation velocity of 33 cm/s is approximately 10 cm, whereas the average distance from the top of the ventricles to the atrioventricular septum is 8 cm. Therefore, the inhomogeneity that is described above cannot exist on the scale of the human heart. Low-conductivity areas that lead to slow propagation of a wave have been introduced in order to adapt this mechanism to the size of the heart. The conductivity values were calculated using the dependence between the conductivity and the velocity of the wave front that is determined using computational methods. Analysis of wave propagation through the boundary between two areas with different conductivities revealed the dependence of the refractory period on the ratio of the conductivities. A transition zone in which the conductivity value changes linearly from the normal value to a lower value has been introduced to attenuate this dependence. As a result, an inhomogeneity that is 12 mm in size that triggers the formation of a circulating wave was modeled.     

Spreading depression in a waking rabbit with the spectral EEG analysis

It was shown that the manifestation of the SD phenomenon in dynamics of the cortical high-frequency gamma activity is rather prominent after bipolar interpretation of the common reference electrode derivations, i.e. when the modeling of the bipolar signal on the base of monopolar EEG recordings is used. The SD wave was accompanied by significant decrease in the power of the EEG gamma band (37-47 Hz) in all observed cortical areas. A curve of decline of gamma activity power had distinct fore and back fronts, so the time of SD appearance in different cortex areas and it's spread succession could be well determined. In the long-term experiments SD waves were characterized by normal (i.e. successive) spread through the cortex mainly in initial three-four experiments. In the subsequent tests latency of SD waves in different cortex areas changed and disturbance of propagation became obvious. Sometimes SD arose rapidly (due 0.5-2 min) on the whole dorsal neocortical surface, when the standard injection of the KCI was done. In the most of experiments the delay of the SD wave appearance was prolonged till 6-9 min or no SD wave occur at some cortical regions. Week epileptiform activity could conduct abnormalities in the SD. In many instances electrophysiological signs of the cortical excitability changes were absent. However the modified spatial SD characteristic and spontaneous occurrence of the repeated. SD waves indicated the increased functional inhomogeneities of the neighboring cortical areas. So, spectral EEG analysis in awake rabbits made it possible to characterize the SD wave both in case of its normal propagation through the cortex and in unusual forms of this reaction.     

Mechanisms of Propagation of Intercellular Calcium Waves in Arterial Smooth Muscle Cells

In rat mesenteric arteries, smooth muscle cells exhibit intercellular calcium waves in response to local phenylephrine stimulation. These waves have a velocity of ∼20 cells/s and a range of ∼80 cells. We analyze these waves in a theoretical model of a population of coupled smooth muscle cells, based on the hypothesis that the wave results from cell membrane depolarization propagation. We study the underlying mechanisms and highlight the importance of voltage-operated channels, calcium-induced calcium release, and chloride channels. Our model is in agreement with experimental observations, and we demonstrate that calcium waves presenting a velocity of ∼20 cells/s can be mediated by electrical coupling. The wave velocity is limited by the time needed for calcium influx through voltage-operated calcium channels and the subsequent calcium-induced calcium release, and not by the speed of the depolarization spreading. The waves are partially regenerated, but have a spatial limit in propagation. Moreover, the model predicts that a refractory period of calcium signaling may significantly affect the wave appearance.     

A Reaction-Diffusion Model of Cholinergic Retinal Waves

Prior to receiving visual stimuli, spontaneous, correlated activity in the retina, called retinal waves, drives activity-dependent developmental programs. Early-stage waves mediated by acetylcholine (ACh) manifest as slow, spreading bursts of action potentials. They are believed to be initiated by the spontaneous firing of Starburst Amacrine Cells (SACs), whose dense, recurrent connectivity then propagates this activity laterally. Their inter-wave interval and shifting wave boundaries are the result of the slow after-hyperpolarization of the SACs creating an evolving mosaic of recruitable and refractory cells, which can and cannot participate in waves, respectively. Recent evidence suggests that cholinergic waves may be modulated by the extracellular concentration of ACh. Here, we construct a simplified, biophysically consistent, reaction-diffusion model of cholinergic retinal waves capable of recapitulating wave dynamics observed in mice retina recordings. The dense, recurrent connectivity of SACs is modeled through local, excitatory coupling occurring via the volume release and diffusion of ACh. In addition to simulation, we are thus able to use non-linear wave theory to connect wave features to underlying physiological parameters, making the model useful in determining appropriate pharmacological manipulations to experimentally produce waves of a prescribed spatiotemporal character. The model is used to determine how ACh mediated connectivity may modulate wave activity, and how parameters such as the spontaneous activation rate and sAHP refractory period contribute to critical wave size variability.     

A Continuum Neuronal Model for the Instigation and Propagation of Cortical Spreading Depression

Cortical spreading depression (CSD) waves can occur in the cortices of various brain structures and are associated with the spread of depression of the electroencephalogram signal. In this paper, we present a continuum neuronal model for the instigation and spreading of CSD. Our model assumes that the brain-cell microenvironment can be treated as a porous medium consisting of extra- and intracellular compartments. The main mechanisms in our model for the transport of ions into and out of neurons are cross-membrane ionic currents and (active) pumps, coupled with diffusion in the extracellular space. To demonstrate the applicability of our model, we have carried out extensive numerical simulations under different initial conditions and inclusion of various mechanisms. Our results show that CSD waves can be instigated by injecting cross-membrane ionic currents or by applying KCl in the extracellular space. Furthermore, the estimated speeds of CSD waves are within the experimentally observed range. Effects of specific ion channels, background ion concentrations, extracellular volume fractions, and cell swelling on the propagation speed of CSD are also investigated.     

Spiral waves of spreading depression in the isolated chicken retina

Existence of the theoretically predicted spiral waves of excitation in intact two-dimensional networks of excitable elements has been experimentally confirmed in the isolated chicken retina. The preparation supports the waves of Leão's spreading depression (SD) the concentric propagation of which from the point of origin can be directly observed as a change of the optical properties of the retinal tissue. The propagation rate of 3.7 mm/min (35°C) decreased to 1.5 mm/min for SD waves elicited during relative refractory period. When a several-mm long segment of the SD wave had been blocked by anodal polarization, the laterally opened ends of the wavefront started to spread after termination of polarization into the previously blocked tissue, gradually turning around and penetrating into the region recovering from the original SD. One or two simultaneously generated spiral waves of SD continued to rotate for several cycles. Spiral SD could also be elicited by punctiform cathodal polarization (1 mA) applied to the SD wave-rear. Since the new SD wave could only spread into the recovering tissue it formed a laterally open wavefront, the free ends of which eventually turned around and started spiral SD. With continued reverberation the nucleus of the spiral SD wave gradually migrated across the retina until it approached an obstacle (e.g., pecten) which stopped further spiral propagation. Spiral SD waves were elicited in 31 retinal preparations and lasted for 4.5 cycles on the average. Average cycle duration was 4.7 min. Spontaneous spiral SD waves were observed in preparations incubated in Mg²⁺-free media. The spiral SD waves in retina are compared with mathematical models of analogous phenomena. It is argued that spiral SD waves probably exist in the cerebral cortex of rats and account for generation of repetitive SD waves sometimes elicited by overlapping stimulation of two cortical regions.     

Self-consistent model of a pulsed air discharge excited by surface waves

A complete self-consistent electrodynamic model of a pulsed gas discharge excited by surface waves is developed. The model allows one to calculate both the initial phase of the discharge front propagation and the parameters of the produced plasma. The spatiotemporal evolution of the electromagnetic field and plasma parameters at the discharge front is investigated for the first time. It is shown that discharge propagation is mainly governed by a breakdown wave in an inhomogeneous electric field at the leading edge of the ionization front. It is found that the effect of the electric field enhancement in the plasma resonance region significantly affects the velocity of the breakdown wave. The results of calculations agree well with experimental data.     

Compression and reflection of visually evoked cortical waves

Neuronal interactions between primary and secondary visual cortical areas are important for visual processing, but the spatiotemporal patterns of the interaction are not well understood. We used voltage-sensitive dye imaging to visualize neuronal activity in rat visual cortex and found visually evoked waves propagating from V1 to other visual areas. A primary wave originated in the monocular area of V1 and was "compressed" when propagating to V2. A reflected wave initiated after compression and propagated backward into V1. The compression occurred at the V1/V2 border, and local GABAA inhibition is important for the compression. The compression/reflection pattern provides a two-phase modulation: V1 is first depolarized by the primary wave, and then V1 and V2 are simultaneously depolarized by the reflected and primary waves, respectively. The compression/reflection pattern only occurred for evoked waves and not for spontaneous waves, suggesting that it is organized by an internal mechanism associated with visual processing.     

Early network activity propagates bidirectionally between hippocampus and cortex

Spontaneous activity in the developing brain helps refine neuronal connections before the arrival of sensory‐driven neuronal activity. In mouse neocortex during the first postnatal week, waves of spontaneous activity originating from pacemaker regions in the septal nucleus and piriform cortex propagate through the neocortex. Using high‐speed Ca²⁺ imaging to resolve the spatiotemporal dynamics of wave propagation in parasagittal mouse brain slices, we show that the hippocampus can act as an additional source of neocortical waves. Some waves that originate in the hippocampus remain restricted to that structure, while others pause at the hippocampus‐neocortex boundary and then propagate into the neocortex. Blocking GABAergic neurotransmission decreases the likelihood of wave propagation into neocortex, whereas blocking glutamatergic neurotransmission eliminates spontaneous and evoked hippocampal waves. A subset of hippocampal and cortical waves trigger Ca²⁺ waves in astrocytic networks after a brief delay. Hippocampal waves accompanied by Ca²⁺ elevation in astrocytes are more likely to propagate into the neocortex. Finally, we show that two structures in our preparation that initiate waves—the hippocampus and the piriform cortex—can be electrically stimulated to initiate propagating waves at lower thresholds than the neocortex, indicating that the intrinsic circuit properties of those regions are responsible for their pacemaker function. © 2015 Wiley Periodicals, Inc. Develop Neurobiol 76: 661–672, 2016     

Vesicular ATP is the predominant cause of intercellular calcium waves in astrocytes

Brain astrocytes signal to each other and neurons. They use changes in their intracellular calcium levels to trigger release of transmitters into the extracellular space. These can then activate receptors on other nearby astrocytes and trigger a propagated calcium wave that can travel several hundred micrometers over a timescale of seconds. A role for endogenous ATP in calcium wave propagation in hippocampal astrocytes has been suggested, but the mechanisms remain incompletely understood. Here we explored how calcium waves arise and directly tested whether endogenously released ATP contributes to astrocyte calcium wave propagation in hippocampal astrocytes. We find that vesicular ATP is the major, if not the sole, determinant of astrocyte calcium wave propagation over distances between approximately 100 and 250 microm, and approximately 15 s from the point of wave initiation. These actions of ATP are mediated by P2Y1 receptors. In contrast, metabotropic glutamate receptors and gap junctions do not contribute significantly to calcium wave propagation. Our data suggest that endogenous extracellular astrocytic ATP can signal over broad spatiotemporal scales.     

Properties of spreading depression during different phases of cyclic excitation of the rat neocortex

Properties of cortical spreading depression were studied during different phases of cyclic excitation developing in the neocortex of rats under the influence of low-frequency electrical stimulation. Waves of spreading depression appeared in the cortex spontaneously or after microinjection of potassium chloride. During each excitation cycle a state preventing the passage and appearance of these waves developed in the region of electrical stimulation. The degree of blocking in other areas of the cortex outside the region of stimulation depended on the distance from the site of electrical stimulation and on generalization of excitatation over the cortex. After the end of the excitation phase, while the current continued to act, the ability of the cortex to conduct the depression wave was restored. In intervals of cyclic excitation the duration of the waves of spreading depression remained on average only half its duration in the absence of stimulation. The time course of development and the character of recovery of depression during the intervals confirm the hypothesis that activation of the potassium-sodium pump may have a role in the blocking mechanism and enable the temporal parameters of this process to be estimated.     

Stimulus-entrained oscillatory activity propagates as waves from area 18 to 17 in cat visual cortex

Previous studies in cat visual cortex reported that area 18 can actively drive neurons in area 17 through cortico-cortical projections. However, the dynamics of such cortico-cortical interaction remains unclear. Here we used multielectrode arrays to examine the spatiotemporal pattern of neuronal activity in cat visual cortex across the 17/18 border. We found that full-field contrast reversal gratings evoked oscillatory wave activity propagating from area 18 to 17. The wave direction was independent of the grating orientation, and could not be accounted for by the spatial distribution of receptive field latencies, suggesting that the waves are largely mediated by intrinsic connections in the cortex. Different from the evoked waves, spontaneous waves propagated along both directions across the 17/18 border. Together, our results suggest that visual stimulation may enhance the flow of information from area 18 to 17.     

Refractory properties of auditory brain-stem responses evoked by electrical stimulation of human cochlear nucleus: evidence of neural generators

In this study of electrically-evoked auditory brain-stem responses (EABRs) elicited by cochlear nucleus stimulation, 3 waves were identified after the initial wave that is directly initiated by the electric stimulus. Varying the rate of periodic stimulation or the interval between pairs of stimuli revealed that the shorter the latency of a wave, the faster it recovered from activation (i.e. shorter refractory period). The slow recovery of the third wave and an accompanying contribution to the second wave could be accounted for by postsynaptic generation in the two medial superior olivary nuclei (MSO); the faster recovery of another contribution to the second wave by generation in an axonal tract bending around the contralateral MSO; and the fastest recovery of the first wave by another axonal pathway having larger axons. Comparison with the relative latencies and spatial distribution of an acoustically-evoked auditory brain-stem response (AABR) indicated that the third wave corresponds to wave V, the second to wave IV (called IVb), and the first to a wave that precedes wave IV (called IVa). The anatomical interpretations for the two later waves of the EABR are consistent with most of the extant data on the neural generators of AABR waves IV and V. Thus, the present data and analysis strengthen the identification of the electrically evoked responses as EABRs and provide a firmer foundation for intra-operative EABR monitoring to assist auditory brain-stem implant placement.     

A Mathematical Model of the Metabolic and Perfusion Effects on Cortical Spreading Depression

Cortical spreading depression (CSD) is a slow-moving ionic and metabolic disturbance that propagates in cortical brain tissue. In addition to massive cellular depolarizations, CSD also involves significant changes in perfusion and metabolism—aspects of CSD that had not been modeled and are important to traumatic brain injury, subarachnoid hemorrhage, stroke, and migraine. In this study, we develop a mathematical model for CSD where we focus on modeling the features essential to understanding the implications of neurovascular coupling during CSD. In our model, the sodium-potassium–ATPase, mainly responsible for ionic homeostasis and active during CSD, operates at a rate that is dependent on the supply of oxygen. The supply of oxygen is determined by modeling blood flow through a lumped vascular tree with an effective local vessel radius that is controlled by the extracellular potassium concentration. We show that during CSD, the metabolic demands of the cortex exceed the physiological limits placed on oxygen delivery, regardless of vascular constriction or dilation. However, vasoconstriction and vasodilation play important roles in the propagation of CSD and its recovery. Our model replicates the qualitative and quantitative behavior of CSD—vasoconstriction, oxygen depletion, extracellular potassium elevation, prolonged depolarization—found in experimental studies. We predict faster, longer duration CSD in vivo than in vitro due to the contribution of the vasculature. Our results also help explain some of the variability of CSD between species and even within the same animal. These results have clinical and translational implications, as they allow for more precise in vitro, in vivo, and in silico exploration of a phenomenon broadly relevant to neurological disease.     

Electric reactive potentials to events of a positive and negative emotional nature

The event-related potentials (ERP) in the frontal and posterior associative cortex in the right and left hemispheres were studied in two different outcomes of tennis television game. These outcomes were "win" and "loss" of the ball, the first serving as a model of positive, the second--of negative emotional reactions. The averaging procedure was synchronized with the moment of ball crossing the racket line. The ERP consisted of four waves: P300, N600, P800, N1000. The most characteristic interhemispheric difference for the win was an increase of N600 wave in the left posterior associative cortex and for the loss--a decrease of P800 wave in the right frontal area. Thus the positive and negative emotional reactions have specific spatiotemporal organization and are based on complicated interaction of cortical areas.     

Nonlinear Gap Junctions Enable Long-Distance Propagation of Pulsating Calcium Waves in Astrocyte Networks

A new paradigm has recently emerged in brain science whereby communications between glial cells and neuron-glia interactions should be considered together with neurons and their networks to understand higher brain functions. In particular, astrocytes, the main type of glial cells in the cortex, have been shown to communicate with neurons and with each other. They are thought to form a gap-junction-coupled syncytium supporting cell-cell communication via propagating Ca²⁺ waves. An identified mode of propagation is based on cytoplasm-to-cytoplasm transport of inositol trisphosphate (IP₃) through gap junctions that locally trigger Ca²⁺ pulses via IP₃-dependent Ca²⁺-induced Ca²⁺ release. It is, however, currently unknown whether this intracellular route is able to support the propagation of long-distance regenerative Ca²⁺ waves or is restricted to short-distance signaling. Furthermore, the influence of the intracellular signaling dynamics on intercellular propagation remains to be understood. In this work, we propose a model of the gap-junctional route for intercellular Ca²⁺ wave propagation in astrocytes. Our model yields two major predictions. First, we show that long-distance regenerative signaling requires nonlinear coupling in the gap junctions. Second, we show that even with nonlinear gap junctions, long-distance regenerative signaling is favored when the internal Ca²⁺ dynamics implements frequency modulation-encoding oscillations with pulsating dynamics, while amplitude modulation-encoding dynamics tends to restrict the propagation range. As a result, spatially heterogeneous molecular properties and/or weak couplings are shown to give rise to rich spatiotemporal dynamics that support complex propagation behaviors. These results shed new light on the mechanisms implicated in the propagation of Ca²⁺ waves across astrocytes and the precise conditions under which glial cells may participate in information processing in the brain.     

Spatiotemporal dynamics of glycolytic waves provides new insights into the interactions between immobilized yeast cells and gels

The immobilization of cells or enzymes is a promising tool for the development of biosensors, yet the interactions between the fixative materials and the cells are not fully understood, especially with respect to their impact on both cell metabolism and cell-to-cell signaling. We show that the spatiotemporal dynamics of waves of metabolic synchronization of yeast cells provides a new criterion to distinguish the effect of different gels on the cellular metabolism, which otherwise could not be detected. Cells from the yeast Saccharomyces carlsbergensis were immobilized into agarose gel, silica gel (TMOS), or a mixture of TMOS and alginate. We compared these immobilized cells with respect to their ability to generate temporal, intracellular oscillations in glycolysis as well as propagating, extracellular synchronization waves. While the temporal dynamics, as measured by the period and the number of oscillatory cycles, was similar for all three immobilized cell populations, significant differences have been observed with respect to the shape of the waves, wave propagation direction and velocity in the three gel matrices used.     

A diffusive SI model with Allee effect and application to FIV

A minimal reaction-diffusion model for the spatiotemporal spread of an infectious disease is considered. The model is motivated by the Feline Immunodeficiency Virus (FIV) which causes AIDS in cat populations. Because the infected period is long compared with the lifespan, the model incorporates the host population growth. Two different types are considered: logistic growth and growth with a strong Allee effect. In the model with logistic growth, the introduced disease propagates in form of a travelling infection wave with a constant asymptotic rate of spread. In the model with Allee effect the spatiotemporal dynamics are more complicated and the disease has considerable impact on the host population spread. Most importantly, there are waves of extinction, which arise when the disease is introduced in the wake of the invading host population. These waves of extinction destabilize locally stable endemic coexistence states. Moreover, spatially restricted epidemics are possible as well as travelling infection pulses that correspond either to fatal epidemics with succeeding host population extinction or to epidemics with recovery of the host population. Generally, the Allee effect induces minimum viable population sizes and critical spatial lengths of the initial distribution. The local stability analysis yields bistability and the phenomenon of transient epidemics within the regime of disease-induced extinction. Sustained oscillations do not exist.