Mathematical approaches to modeling of cortical spreading depression

Mathematical Psychiatry: On Cortical Spreading Depression-A Review

The concept of migraine with aura (MwA) is a widespread condition that can affect up to 30% of migraine patients and manifests itself as a temporary visual illusion followed by a prolonged headache. It was initially pitched as a neurological disease, and observed that the spread of accompanying electrophysiological waves as part of the condition, which came to be known as cortical spreading depression (CSD). A strong theoretical basis for a link between MwA and CSD has eventually led to knowledge of the dynamics between the pair. In addition to experiment-based observations, mathematical models make an important contribution towards a numerical means of expressing codependent neural-scale manifestations. This provides alternate means of understanding and observing the phenomena while helping to visualize the links between the variables and their magnitude in contributing towards the emanation and dynamic pulsing of the condition. A number of biophysical mechanisms are believed to contribute to the MwA-CSD, spanning ion diffusion, ionic currents of membranes, osmosis, spatial buffering, neurotransmission, gap junctions, metabolic pumping, and synapse connections. As part of this review study, the various mathematical models for the description of the condition are expressed, reviewed, and contrasted, all of which vary in their depth, perspective, and level of information presented. Subsequent to this, the review looked into links between electrophysiological data-driven manifestations from measurements such as EEG and fMRI. While concluding remarks forged a structured pathway in the area on sub-themes that need to be investigated in order to strengthen and robustify the existing models, they include an accounting for inter-personal variability in models, sex and hormonal factors, and age groups, i.e., pediatrics vs. adults.

Quantifying the relationship between spreading depolarization and perivascular cerebrospinal fluid flow

Recent studies have linked spreading depolarization (SD, an electro-chemical wave in the brain following stroke, migraine, traumatic brain injury, and more) with increase in cerebrospinal fluid (CSF) flow through the perivascular spaces (PVSs, annular channels lining the brain vasculature). We develop a novel computational model that couples SD and CSF flow. We first use high order numerical simulations to solve a system of physiologically realistic reaction-diffusion equations which govern the spatiotemporal dynamics of ions in the extracellular and intracellular spaces of the brain cortex during SD. We then couple the SD wave with a 1D CSF flow model that captures the change in cross-sectional area, pressure, and volume flow rate through the PVSs. The coupling is modelled using an empirical relationship between the excess potassium ion concentration in the extracellular space following SD and the vessel radius. We find that the CSF volumetric flow rate depends intricately on the length and width of the PVS, as well as the vessel radius and the angle of incidence of the SD wave. We derive analytical expressions for pressure and volumetric flow rates of CSF through the PVS for a given SD wave and quantify CSF flow variations when two SD waves collide. Our numerical approach is very general and could be extended in the future to obtain novel, quantitative insights into how CSF flow in the brain couples with slow waves, functional hyperemia, seizures, or externally applied neural stimulations.

A mathematical model for persistent post-CSD vasoconstriction

Cortical spreading depression (CSD) is the propagation of a relatively slow wave in cortical brain tissue that is linked to a number of pathological conditions such as stroke and migraine. Most of the existing literature investigates the dynamics of short term phenomena such as the depolarization and repolarization of membrane potentials or large ion shifts. Here, we focus on the clinically-relevant hour-long state of neurovascular malfunction in the wake of CSDs. This dysfunctional state involves widespread vasoconstriction and a general disruption of neurovascular coupling. We demonstrate, using a mathematical model, that dissolution of calcium that has aggregated within the mitochondria of vascular smooth muscle cells can drive an hour-long disruption. We model the rate of calcium clearance as well as the dynamical implications on overall blood flow. Based on reaction stoichiometry, we quantify a possible impact of calcium phosphate dissolution on the maintenance of F0F1-ATP synthase activity.

Patient-specific computational modeling of cortical spreading depression via diffusion tensor imaging

Cortical spreading depression, a depolarization wave originating in the visual cortex and traveling towards the frontal lobe, is commonly accepted as a correlate of migraine visual aura. As of today, little is known about the mechanisms that can trigger or stop such phenomenon. However, the complex and highly individual characteristics of the brain cortex suggest that the geometry might have a significant impact in supporting or contrasting the propagation of cortical spreading depression. Accurate patient-specific computational models are fundamental to cope with the high variability in cortical geometries among individuals, but also with the conduction anisotropy induced in a given cortex by the complex neuronal organisation in the grey matter. In this paper, we integrate a distributed model for extracellular potassium concentration with patient-specific diffusivity tensors derived locally from diffusion tensor imaging data.

Aquaporin-4 regulates the velocity and frequency of cortical spreading depression in mice

The astrocyte water channel aquaporin-4 (AQP4) regulates extracellular space (ECS) K(+) concentration ([K(+)]e) and volume dynamics following neuronal activation. Here, we investigated how AQP4-mediated changes in [K(+)]e and ECS volume affect the velocity, frequency, and amplitude of cortical spreading depression (CSD) depolarizations produced by surface KCl application in wild-type (AQP4(+/+)) and AQP4-deficient (AQP4(-/-)) mice. In contrast to initial expectations, both the velocity and the frequency of CSD were significantly reduced in AQP4(-/-) mice when compared with AQP4(+/+) mice, by 22% and 32%, respectively. Measurement of [K(+)]e with K(+)-selective microelectrodes demonstrated an increase to ∼35 mM during spreading depolarizations in both AQP4(+/+) and AQP4(-/-) mice, but the rates of [K(+)]e increase (3.5 vs. 1.5 mM/s) and reuptake (t1/2 33 vs. 61 s) were significantly reduced in AQP4(-/-) mice. ECS volume fraction measured by tetramethylammonium iontophoresis was greatly reduced during depolarizations from 0.18 to 0.053 in AQP4(+/+) mice, and 0.23 to 0.063 in AQP4(-/-) mice. Analysis of the experimental data using a mathematical model of CSD propagation suggested that the reduced velocity of CSD depolarizations in AQP4(-/-) mice was primarily a consequence of the slowed increase in [K(+)]e during neuronal depolarization. These results demonstrate that AQP4 effects on [K(+)]e and ECS volume dynamics accelerate CSD propagation.

Computational modeling of neurostimulation in brain diseases

Neurostimulation as a therapeutic tool has been developed and used for a range of different diseases such as Parkinson's disease, epilepsy, and migraine. However, it is not known why the efficacy of the stimulation varies dramatically across patients or why some patients suffer from severe side effects. This is largely due to the lack of mechanistic understanding of neurostimulation. Hence, theoretical computational approaches to address this issue are in demand. This chapter provides a review of mechanistic computational modeling of brain stimulation. In particular, we will focus on brain diseases, where mechanistic models (e.g., neural population models or detailed neuronal models) have been used to bridge the gap between cellular-level processes of affected neural circuits and the symptomatic expression of disease dynamics. We show how such models have been, and can be, used to investigate the effects of neurostimulation in the diseased brain. We argue that these models are crucial for the mechanistic understanding of the effect of stimulation, allowing for a rational design of stimulation protocols. Based on mechanistic models, we argue that the development of closed-loop stimulation is essential in order to avoid inference with healthy ongoing brain activity. Furthermore, patient-specific data, such as neuroanatomic information and connectivity profiles obtainable from neuroimaging, can be readily incorporated to address the clinical issue of variability in efficacy between subjects. We conclude that mechanistic computational models can and should play a key role in the rational design of effective, fully integrated, patient-specific therapeutic brain stimulation.

Refractory period modulates the spatiotemporal evolution of cortical spreading depression: a computational study

Cortical spreading depression (CSD) is a pathophysiological phenomenon, which underlies some neurological disorders, such as migraine and stroke, but its mechanisms are still not completely understood. One of the striking facts is that the spatiotemporal evolution of CSD wave is varying. Observations in experiments reveal that a CSD wave may propagate through the entire cortex, or just bypass some areas of the cortex. In this paper, we have applied a 2D reaction-diffusion equation with recovery term to study the spatiotemporal evolution of CSD. By modulating the recovery rate from CSD in the modeled cortex, CSD waves with different spatiotemporal evolutions, either bypassing some areas or propagating slowly in these areas, were present. Moreover, spiral CSD waves could also be induced in case of the transiently altered recovery rate, i.e. block release from the absolute refractory period. These results suggest that the refractory period contributes to the different propagation patterns of CSD, which may help to interpret the mechanisms of CSD propagation.

Introduction to focus issue: rhythms and dynamic transitions in neurological disease: modeling, computation, and experiment

Rhythmic neuronal oscillations across a broad range of frequencies, as well as spatiotemporal phenomena, such as waves and bumps, have been observed in various areas of the brain and proposed as critical to brain function. While there is a long and distinguished history of studying rhythms in nerve cells and neuronal networks in healthy organisms, the association and analysis of rhythms to diseases are more recent developments. Indeed, it is now thought that certain aspects of diseases of the nervous system, such as epilepsy, schizophrenia, Parkinson's, and sleep disorders, are associated with transitions or disruptions of neurological rhythms. This focus issue brings together articles presenting modeling, computational, analytical, and experimental perspectives about rhythms and dynamic transitions between them that are associated to various diseases.