Models of cortical spreading depression

• Markus A. Dahlem, Institute of Theoretical Physics, TU Berlin, Germany

Dr. Markus A. Dahlem accepted the invitation on 25 February 2009 (self-imposed deadline: 25 September 2009).

Models of cortical spreading depression refers to mathematical descriptions of cortical spreading depression. Cortical spreading depression is a massive but temporary perturbation in the cortical ionic homeostasis leading to a depression of neuronal activity that spreads (hence the name) through the cortex and other gray matter regions in the brain. Cortical spreading depression is an emergent phenomenon arising from the interaction of many different transmission phenomena in the cortex.

The pathophysiological basis for modeling SD

Figure 1: Flow chart of feedback loops and pathways in SD.

To understand the essential features of cortical spreading depression presented in this article, an elementary knowledge of electrophysiology is necessary, in particular, for the detailed cellular models presented, and also an elementary knowledge of thermodynamics far from equilibrium, in particular for the macroscopic models. Note that there are also experimental models of this phenomenon referring to a particular experimental procedure, animal species, and/or brain region.

Terminology

In this article, SD is used as the abbreviation for spreading depression. Omitting the "C" is advisable as some experimental data are not obtained from the cortex. Moreover, this phenomenon is in recent years frequently called spreading depolarization, a term with the same abbreviation but that reflects what is thought to be a main step in the mechanism, namely the depolarization of the membrane potential.

Local aspects of SD

During SD a massive redistribution of ions across cell membranes peaks after several seconds in a nearly complete neuronal depolarization with a depression of neural activity, followed by a much slower recovery process taking up to minutes during which ion gradients are re-established towards their physiological values. This maximal ionic perturbation clearly distinguishes SD from all other brain states such as epileptic seizure activity, functional activation or the physiological resting state. The sequence of ionic perturbation and its recovery spreads at a pace of about 3~mm/min over cortical regions. Cortical spreading depression is closely related to migraine and stroke.

Models of cortical spreading depression can be distinguished by their level of detail and complexity. In a nutshell, cellular models of SD are based on a Hodgkin-Huxley type description, that is, they use Kirchhoff's second to law the describe electrophysiological events across the neuronal cell membrane. But unlike the Hodgkin-Huxley model of action potentials, models of SD must also consider surrounding compartments to account for changes in the intra- and extracelluar ion concentrations. Under physiological conditions, these chemical gradients are rather stable batteries and can therefore be considered as constant. By contrast, the characteristic event in SD is the break-down of the ionic homeostasis. To keep track of the changes in the ion concentration three phenomena are usually accounted for in models of SD:

• the Goldman-Hodgkin-Katz (GHK) current equation
• ion exchange pumps
• glial buffering

Spatial aspects of SD

Obtaining the spatial continuum limit of such discrete microscopic cellular models of SD is by no means straightforward, although efforts can be made in this direction. The continuum limit is necessary to model the spread, which can under conditions of migraine and stroke extend over several centimeters in the human cortex. Continuum limit requires appropriate conditions on the spatial coupling, that is, on the communication pathways between cortical cells. Communication between neurons is usually classified in the cortex by two schemes:

• wiring transmission
• volume transmission

Volume transmission is mainly characterized by diffusion of chemical signals in the extracellular space. Wiring transmission is based on the structural and functional cortical connectivity, and other intercellular communication occurring through a well-defined network.

An alternative to the continuum limit of a cellular model is a macroscopic model utilizing a top-down approach. In this case, the mechanism of SD is described as a generic reaction-diffusion system. This is the other end of the wide spectrum of SD models. Furthermore, there are cellular automaton models of SD and kinematic descriptions of the SD wave front, but these models play a minor role to date, though there is some historical interest. Finally, there are hybrid models, that is, models using a macroscopic reaction-diffusion description in combination with neural network models. Such a model model was first suggested to explain visual hallucinations caused by SD during the course of a migraine attack (Reggia and Montgomery, 1996), but recent developments focus the attention now on the neural circuity as an augmented transmission scheme that provides weak nonlocal coupling and thereby it may modulate cortical susceptibility to SD.

Early modeling attempts

Figure 2: First page from Wiener and Rosenblueth paper (1941). Copy with annotations made by the late Art Winfree.

Figure 3: Spiral-shaped retinal SD, overlay.

In 1946, Norbert Wiener and Arturo Rosenblueth proposed a mathematical framework to describe wave patterns explaining certain types of cardiac arrhythmia. The authors made also the connection to brain dynamics, stating that ``Nervous elements and cardiac [...] fibers are excitable. [...] The laws which apply to the muscle fibers are also applicable to the nerve fibers``. Wiener and Rosenblueth knew very well, in fact, that waves similar to the those in cardiac arrhythmia occur in the cortex. Arturo Rosenblueth, in collaboration with Hallowell Davis, supervised a few years earlier a young Brazilian PhD student, Aristides A. P. Leao who was working in the department of Physiology at Harvard Medical School and discovered the phenomenon of SD in 1944.

In Figure 2, the title page of this seminal paper by Wiener and Rosenblueth is shown with many annotations made 1970 by Art Winfree, in particular, mentioning rotors, that is, a special concept of re-entrant waves that are not pinned by an anatomical block but rotate freely. These waves are also called spiral waves. Winfree was a leading theoretical biologist who made major contributions to the field of biological oscillations. In 1972, he published a paper on Spiral Waves of Chemical Activity observed in the Belousov-Zhabotinsky (BZ) reaction. Three years later, Reshodko and Bures (1975) published the first computer simulation of re-entrant SD waves in a network of cell automata.

These early modeling attempts that focus on the two-dimensional propagation pattern of SD on a centimeter scale have recently regained much interest, for monitoring the human brain provided evidence of re-entrant SD patterns during stroke. Stroke outcome could critically depend on SD as these events pose dramatic metabolic stress to the tissue. Whether or not the loss of potentially salvageable tissue, i.e., tissue at risk of infarction, is increased by the number of SD is a current research focus. Computational model of SD complement traditional research methodologies (Ruppin et al. 1999).

Biophysical ionic-based model

The basic dynamical equations below are formulated for monovalent cations \(Na^+\) and \(K^+\ .\) This modeling scheme describes the basic framework, which can easily be extended. For example, fluxes of \(Cl^{-}\) and \(Ca^{2+}\) can likewise be computed and their feedback effect on the neuron can be taken into account. Furthermore, for both \(Na^+\) and \(K^+\) various currents can be implemented. In particular, for \(Na^+\) a fast transient sodium current and a persistent sodium current, and for \(K^+\) an inward rectifying potassium current and a delayed outward rectifier potassium current are the main currents types of these monovalent cations. In the follwoing, we start, however, with a scheme having a single, not further specified current for each of the two cations \(Na^+\) and \(K^+\ .\)

Kirchhoff's current law states that the sum of all currents must equal zero. Chosing the displacement current $C \frac{\partial V}{\partial t}$ (due to the membrane capacitance \(C\)) as the left hand side leads to the main equation:

\( C \frac{\partial V}{\partial t} = -m I_{Na} - n I_{K} - I^{pump}_{K}(V) - I^{pump}_{Na}(V) \)

Changes in the membrane potential \(V\) are due to two types of transmembrane currents:

\( I_{ion}= V\, \alpha F\, P_{ion} \frac{[{ion}]_i - [ion]_o e^{-\alpha V}}{1-e^{-\alpha V}} \) \( I^{pump}_{ion}(V)= \beta_{ion} I_{max} \left( 1+\frac{K_{m_K}}{[K]_o} \right)^{-2} \left( 1+\frac{K_{m_{Na}}}{[Na]_i} \right)^{-3} \) that is:

(i) Goldman-Hodgkin-Katz (GHK) ionic currents \(I_{ion}\ ,\) where \(ion \in \{Na^+, K^+\}\ ,\) \(P_{ion}\) is the membrane permeability, and the parameter $\alpha=\frac{F}{RT}\(, with \)F\( Faraday's constant, \)R\( the gas constant, \)T$ the absolute temperature,

(ii) pump currents \(I^{pump}_{ion}(V)\) (Eq.~(\ref{eq:wil5})), with \(K_{m_{ion}}\) being the ion affinities, and a maximum pump current \(I_{max}\) that under steady state (resting) conditions compensates the ion leak GHK currents, and the constant $\beta_{ion}$ is \(-2\) and \(3\) for ion \(K^+\) and \(Na^+\ ,\) respectively.

The voltage-dependent conductances \(n\) and \(m\) of \(K^+\) and \(Na^+\) are modeled by

\( \frac{\partial n}{\partial t}= 1/\tau_n(n_{\infty} - n) \) \( \frac{\partial m}{\partial t}= 1/\tau_m(m_{\infty} - m) \)

On the relevant time scale of SD, the dynamics may be approximated by their steady state dynamics, \(m_\infty(V)\) and \(n_\infty(V)\ ,\) respectively.

In this model, the membrane currents are carried by massive ionic changes (in contrast to normal brain activity), and this has to be taken into account as an actual change in the ion's intra- and extracellular concentrations, modeled by the rate equation of \([ion]_i\) and \([ion]_o\ ,\) respectively

\( \frac{\partial [ion]_o}{\partial} = \frac{I_{ion} A}{F V\!ol_o} + D_{ion} \nabla^2 [ion]_o \) \( \frac{\partial [ion]_i}{\partial t} = \frac{I_{ion} A}{F V\!ol_i} \)

This makes it necessary to define the neural membrane surface area \(A\) and an intra- and extracellular space volume, denoted \({V\!ol_i}\) and \({V\!ol_o}\ ,\) respectively. The interstitial volume fraction \(\frac{V\!ol_o}{V\!ol_i}\) will be a fixed at \(\frac{V\!ol_o}{V\!ol_i}=0.15\ ,\) although this might need to be changed at a later stage, because osmotic changes could play an important role in SD. This set of equations can also be extended to include further effects, like a first-order buffering scheme simulating an effective glial \(K^+\) uptake system.

Our aim is to use the simplest plausible approximation that is consistent with the diversity of dynamical behavior observed and couple the effect of an external electrical field as a control signal to the SD model.

Macroscopic reaction-diffusion models

Neurons and also glial cells in brain tissue are in a non-equilibrium state stationary state. During SD, the brain tissue approaches its equilibrium state, which would eventually lead to the loss of this tissue, if the recovery to the non-equilibrium state stationary is not achieved within minutes. In a nutshell for physicists, SD is thus a pulse-like solitary wave in which the tissue temporarily approaches nearly an equilibrium state and is then driven back far from it.

Hybrid models

Reaction-diffusion SD model forcing a neural network

Figure 4: A model of hallucinatory zigzag caused by SD wave .

Figure 5: Simulation.

In the mid 1990s, Reggia and Montgomery (1996) proposed a SD model to explain the hallucinatory zigzag pattern that is thought to be caused by the initial excitation phase of SD. To test this, at that time controversial, hypothesis, they extended a reaction-diffusion (RD) model based on potassium dynamics, which was similar to the Hodgkin-Grastein model only with a quartic polynomial, by including a second dynamic variable describing a recovery phase, identical to the FitzHugh-Nagumo inhibitor rate equation, and connected this RD model to a neural network. The neural network was initially not developed to study migraine aura but used to study cortical dynamics and sensory map reorganization.

The distinguishing and novel feature of the Reggia and Montgomery (1996) SD model is that it connected two types of models to become a hybrid model. The key result of their simulations is that on the leading edge of the potassium wave, the elsewhere largely uniform neural activity was replaced by a pattern of small, irregular patches and lines of highly active elements. In 1971, Richards suggested in a Scientific American article about visual migraine aura that the zigzag seen in a visual aura is caused by the spatial layout of a specific type of neurons in visual cortex. Therefore, SD by evoking migraine hallucinations provides a route to the neural mechanisms of normal brain function.

It was argued that the approaching SD wave initially affects cortical cells located in a special layer of the cortex (IVc) that possess the highest spontaneous activity. In layer IVc, the afferent inputs to the cortex terminate, and cells located there posses a concentric circular receptive field structure. Cells in the surrounding laminae possess fairly low spontaneous discharge rate, and presumably are strongly inhibited by intracortical inhibition. Therefore, those cells which converge to orientations which are perpendicular to the SD front will be inhibited first. Their excitatory input to the surrounding simple cells will be attenuated, and the activity of these surrounding cells will be depressed. Correspondingly, their neighbors on either side will be disinhibited, and will experience an elevation in firing rate. These columns correspond to orientations which are in the range of 30° to 60° from perpendicular to the advancing front, and thus would be perceived as the jagged fortification.

Neural network models have become sophisticated enough to constrain and validate possible an SD attack underlying cortical circuitry. Simulations with SD driven population codes interpreting cortical feature maps an transforming them into a percept in the visual field have lead to simulating of the aura that allow the patient to virtually re-experience their disorders (see Figure 4). Preliminary results indicate that such networks provide realistic simulations that can be matched by patients against their visual symptoms.

References

• Reggia, J. A. and Montgomery, D. , A computational model of visual hallucinations in migraine, Comput. Biol. Med. 26, 133-139 (1996).

• Ruppin, E. , Revett, K. , Ofer, E. , Goodall, S. and Reggia, J. A. , Penumbral tissue damage following acute stroke: a computational investigation, Prog. Brain Res. 121, 243-255 (1999)

• Reshodko, L. V. and Bures J., Computer simulation of reverberating spreading depression in a network of cell automata, Biol. Cybern. 18, 181-189 (1975).
• Winfree, A. T., Spiral Waves of Chemical Activity, Science 175, 634-36 (1972).

Internal references

• Gregoire Nicolis and Anne De Wit (2007) Reaction-diffusion systems. Scholarpedia, 2(9):1475.
• Eugene M. Izhikevich and Richard FitzHugh (2006) FitzHugh-Nagumo model. Scholarpedia, 1(9):1349.

Sponsored by: Eugene M. Izhikevich, Editor-in-Chief of Scholarpedia, the peer-reviewed open-access encyclopedia