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

Abstract

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.

Graphical abstract

Keywords

cortical spreading depression / time-varying / relative refractory period / computational study

INTRODUCTION

Cortical spreading depression (CSD) was first observed in the cerebral cortex of rabbits by Leão in 1944 [ 1, 2]. CSD is a pathophysiological phenomenon characterized by a transient suppression of neuronal activity, failure of ion homeostasis, a negative direct current (DC) potential shift and the depression of electrocorticogram [ 3– 6]. During CSD, oxygen consumption and local cerebral blood flow will increase [ 7], and the cerebral vessel will change [ 8]. CSD is suspected to be involved in migraine [ 9– 11], and have an underlying role in epilepsy [ 6, 12, 13] and stroke [ 14, 15]. Different kinds of methods, including mechanical injury, chemical or anoxic stimuli [ 16– 20], have been used to initiate CSD in animal brain to study the mechanisms of CSD. However, the mechanisms of CSD are still not completely clear [ 6, 21, 22].

CSD has been observed with electrophyiological, ionic and optical changes spreading in cortex [ 23]. Characterizing propagation dynamics of CSD may help to understand its unclear mechanisms. Here, we are focusing on the typical ionic changes of potassium and calcium. Previously, two types of propagation events had been recorded during the repetitive CSD induced by KCl [ 18, 24]. The primary event was the first wave propagating through the cortex, while the secondary events were the succeeding waves. These two kinds of events had similar duration, but some of the propagation patterns such as amplitude and velocity were different [ 18, 24, 25]. This implied that the primary event can modulate the following events, but the nature of the modulation is still not known [ 26]. Our work has demonstrated that the elicitation interval of CSD waves has effect on the spatiotemporal evolution of CSD [ 27, 28]. Independent on the stimulation of pinprick or KCl, a short interval of the current CSD to the last CSD no more than 4 min would induce the current CSD be partially propagated [ 28]. Thus, CSD could not be an “all or none” phenomenon [ 25]. This challenged the traditional conception.

Usually, the different regional manifestation of CSD was thought as the different cytoarchitecture and glia/neuron ratio [ 5, 6, 29, 30]. However, the temporal variation of propagation patterns of successive CSD waves in the same region could not be entirely accounted by the cytoarchitecture. In fact, the modulation of the refractory period of CSD was found involved in the variation [ 31– 33]. Both the experimental and theoretical work addressed the mechanisms of gap junctions, glial cells (astrocytes) and ion channels. The simulation results revealed the refractoriness conditions of CSD (i.e., the slowly inactivating inward currents overcome the outward ones) [ 32, 34]. In order to bridge the gap between the macroscopic and microscopic level of CSD, this study paid more attention to study the variation of CSD patterns under long-lasting and transient stimulation conditions. This would be helpful to control the stimulation condition so that the uniform CSD wave is induced for practical requirement [ 35]. Correspondingly, a phenomenological model was introduced in this paper.

RESULTS

Dynamics of ionic concentrations during a single CSD wave

After a transient injection of KCl with concentration of 20 mM, only one CSD was evoked in the modeling system. The changes of ionic concentrations accompanied CSD were shown in Figure 1: [K⁺]_o rose up to 60 mM, [Ca²⁺]_a up to 0.17 mM, [Ca²⁺]_n up to 0.13 mM, and [Ca²⁺]_o down to 0.33 mM. All these 4 simulated ionic changes were consistent with the experimental records [ 5, 36]. The duration of [K⁺]_o change was about 100 s (measured at half of the maximum amplitude), then it returned to the resting state within 3 min. We focused on three sites (3 mm, 7 mm, 11 mm from the injection center respectively), acting as probes in the simulation to calculate the velocity of CSD. The CSD propagated through the cortex at a speed of 3.7 mm/min in this model, which was similar to the experimental data [ 5, 6]. In Figure 2, a continuous KCl injection lasting an hour could bring a series of time-varying CSD waves in the region of interest. This phenomenon mimicked the experiments [ 18, 25, 27]. Both the transient and sustained stimulation results validated the current model.

Temporal evolution of repetitive CSD waves

The modeling results of repetitive CSD waves also reproduced that the amplitude of the primary event was larger than the average of the secondary events [ 26]. The first wave of [K⁺]_o could rise nearly to 60 mM, while the average peak value of the following waves in an hour was just about 38 mM, decreasing by 22 mM. The profile of the primary wave during the repetitive CSD waves was similar to that of the single one in our modeling [ 27, 37]. The amplitudes of the secondary events did not decrease monotonously but varied with time (Figure 2). The amplitudes of the several subsequent waves were always smaller than that of the primary one. In addition, the primary CSD wave had a speed of 3.7 mm/min and the average speed of the secondary CSD waves was just 1.7 mm/min. But the duration of the primary wave was similar to that of each succeeding CSD wave [ 24]. With careful inspection and calculation, a longer time interval after the last CSD would give birth to the following CSD with larger amplitude and faster velocity in this model [ 31, 32]. In fact, the primary wave was found propagating through the entire cortex but the following waves might bypass some area of the cortex in experiments [ 18, 26– 28]. This suggested the temporal evolution of CSD was related with spatial evolution: a CSD wave owning large amplitude could propagate more wide area of cortex. The spatiotemporal evolution of CSD waves was related with the time interval between CSD waves [ 27]. This modeling work verified this point.

The dominating factor of varying CSD waves

Here, we attempted to seek which factor is dominated to the varying CSD waves in the modeling work. All the six variables during the first two CSD waves were plotted in Figure 3. Even all of the ions involved in this model returned to the resting state, but R was still in the excitement. It hinted that R might have a long-lasting influence on the propagation dynamics of CSD waves. The simulation experiment was performed to validate our conjecture in Figure 4. In the case that R did not recover to the resting state, the following CSD would not have the same profile as the previous one (Figure 4A). More further away for R returning to the resting state made a larger difference between the following CSD and the previous one. When R went back to the normal nearly, the following CSD would have the same profile as the previous one (Figure 4B). Since R represents as the recovery mechanism, R could be a quantitative variable for the relative refractory period.

Relative refractory period and variation of CSD waves

Two events of transient injection of KCl with different time intervals during the recovery period were modeled to observe the influence of refractory period on the propagation dynamics of CSD waves. The first KCl injection happened soon after the simulation. The second stimulation intervals were 240 s, 300 s, 360 s, 420 s, 480 s and 600 s, respectively. It appeared that if the stimulation interval was too short [ 5], CSD could not be evoked (Figure 5A). The reason was that the stimulation was made during the absolute refractory period. This also indicated that the change of variable R covered the absolute and relative refractory period. An unexpected phenomenon was observed that a later injection of KCl could induce an earlier appearance of CSD (Figure 5B–E). The CSD occurring later had a little faster velocity and a little larger amplitude (Data are list in Table 1). When the stimulation interval was long enough, for example, 600 s in this model, the secondary events had nearly the same amplitude and velocity as the primary event (Figure 5F). Ten minutes should be considered as the time that the tissue needs to recover from CSD. It seemed that long interval could reduce the difference among the repetitive CSD waves.

In the above work, there seemed to be a positive correlation between the amplitude and the velocity during the repetitive CSD waves. Here, a CSD was evoked by transient stimulation at the beginning of the simulation. Another succeeding stimulation injection was carried out at different time points from about 300 s to 1000 s to the first stimulation. Within 290 s, we did not give any stimulation because no CSD can be induced during the absolute refractory period. For each trial, there were two CSD waves induced. The amplitude and velocity of the second CSD wave were recorded at a location 7 mm from the injection center. The time interval of the peak values of the two CSD waves called peak interval was also recorded there. In Figure 6, it appeared that the amplitude of CSD had the same trend as the velocity [ 32]. The larger the amplitude was, the faster the velocity. A decline at the beginning of the curves was noticed. It suggested that under a certain time frame during the relative refractory period, an early injection could evoke a CSD having a little faster velocity as well as a little larger amplitude. From 350 s to 600 s, there was nearly a linear relationship between the amplitude and velocity of CSD. About 600 s later, the amplitude and the velocity changed little, and reached a stable state. CSD evoked at that stable state had almost the same profile as the first CSD wave. Here it suggested that it cost about 10 minutes for the tissue to recover from CSD completely.

In Figure 7, the second CSD could not be observed at the location 7 mm from the injection center until 400 s after the first CSD was observed. As peak interval increased, the amplitude of CSD got larger and the velocity became faster. Also, the amplitude and velocity would reach a stable state after 600 s. It was found that it was not a linear relationship between stimulation interval and peak interval in Figure 8. During the relative refractory period, a short stimulation interval might lead to a long peak interval. The longer peak interval meant that the second CSD occurred much later. So, a testable prediction came into being that under a certain time window during the relative refractory period, an earlier injection could evoke a later appearance of CSD, and the CSD occurring later had a little faster velocity as well as a little larger amplitude. This demonstrated that haste makes waste.

DISCUSSION

Since the clinical relevance of CSD, there has been extensive interest in both experimental and computational aspects of CSD [ 6, 13, 32, 38– 42]. This paper aimed to simulate the variation of repetitive CSD waves found in experiments with a computational study [ 18, 25, 27, 28]. It was different from some previous theoretical work that addressed the characteristics of a solitary CSD wave [ 43– 45]. Many experiments showed the inhomogeneous spatiotemporal evolution of successive CSD waves, but the modeling work on this issue was still few [ 32, 33]. The present modeling has been successful in simulating the single or successive CSD waves. Also, this work has shown that the variation of repetitive CSD waves is related with the relative refractory period. Since the current model was only one spatial dimension, the variation of CSD was mainly shown as temporal ionic changes. The spatiotemporal dynamics had been qualitatively studied in our previous work [ 33].

In this study, R represented as the recovery mechanism, especially standing for relative refractory period. It was shown that R lasts a longer duration than the extracellular ion disturbance. This suggested that even all of ions go back to the resting state, the tissue is not completely recovered from the last CSD wave with non-resting R value. This was in accordance with the previous study about the refractoriness [ 31, 32]. So, it was easy to control the variable of R to modulate the successive CSD waves. By adjusting some parameters, the variable of R were also found related with the amplitude and frequency of CSD waves (Results not shown). This was consistent with the experimental observation with pharmacological blockers [ 31, 46].

An interesting result of this model was that an earlier stimulation during relative refractory period could induce a later appearance of CSD, i.e., “haste makes waste”. This should be a testable prediction in the future experiments. The amplitude and velocity of CSD wave were influenced by the stimulation interval. Further, they had a positive correlation with the peak time interval (i.e., occurrence interval). The amplitude had the same trend as the velocity. This point was similar with previous studies [ 31, 32, 44]. But the previous work did not distinguish the stimulation interval and occurrence interval (i.e., peak interval in Figures 7 and 8). The computational results showed that the amplitude and velocity of successive CSD waves were varying. It inferred that CSD is not necessarily an “all or none” phenomenon. Noticed from the experimental results, the CSD wave had larger amplitude and higher velocity would spread to more wide area of cortex [ 18, 25, 27, 28]. If the uniform CSD wave was required under practical consideration [ 35], a long enough recovery time about 10 minutes were suggested here.

This model could be used to study some other factors that could influence the propagation dynamics of the repetitive CSD waves, such as gap junction and astrocytes [ 47, 48]. However, the current work was mainly a phenomenological model without accurate knowledge of cytoarchitecture and ion channels of cortex. This was not a cellular level model of CSD. The current model had limitations to reflect the individual neurons and glial cells. Neither the function of sodium channels nor the changes of sodium ions were not included yet. In fact, we paid more attention to the rational phenomenal results than the completeness of the model formation. This work suggested that the refractory period is critical for preventing undesirable CSD waves. To the best of our knowledge, the philosophy of “haste makes waste” is first predicted from the proposed model. Taken together, this macroscopic level study of CSD had been proved helpful to modulate and understand the spatiotemporal evolution of CSD waves.

METHODS

The conceptual model

We incorporated part of the model from Chapuisat [ 49] and Tuckwell [ 38], forming a one dimensional continuum model presented here (Figure 9A). Three compartments in the cortical tissue (gray matter) were considered. They were neurons, glial cells and the extracellular space. Here, blood vessels were not included in this model. Since astrocytes play an important role in CSD [ 39, 47], we took glial cells as astrocytes to simplify the model. The model consists of two ionic species, the potassium and the calcium. [K⁺]_o, [Ca²⁺]_a, [Ca²⁺]_n and [Ca²⁺]_o respectively represent as the extracellular potassium concentration, the astrocytic calcium concentration, the neuronal calcium concentration, and the extracellular calcium concentration. We did not take the change of the volume of the three compartments into account during CSD. The volume of each compartment was kept as constant. We considered the fraction of extracellular space as 0.2 [ 50], and the fractions of neurons and astrocytes are 0.07, 0.73 respectively [ 33]. Our model complies with such following physiological processes (Figure 9B). The stimulation is injection of KCl solution to induce an initial increase of [K⁺]_o. After injecting KCl, the potassium concentration in the extracellular space [K⁺]_o rises. The increase of [K⁺]_o causes all the surrounding cells to depolarize. This makes the voltage-gated calcium channels in astrocytes open, leading calcium enter astrocytes and [Ca²⁺]_a increase. Calcium in the astrocytes propagates through gap junctions between astrocytes. Some of calcium enters neurons, making [Ca²⁺]_n increase and open calcium-dependent potassium channels. Thus, the potassium in neurons goes out to the extracellular space, which is helpful to the increase of [K⁺]_o. When the [K⁺]_o is higher than certain threshold, CSD may be induced and propagate. During CSD, the ion homeostasis is interrupted and undergoing a large fluctuation. A few minutes later, the cortical tissue returns to the resting state without successive stimulation.

The equations and the parameters

The model is made up of two partial differential equations, three ordinary differential equations and an algebraic equation. The Equations 1–4 control the ionic movements, and the variables Ca_a, Ca_n, Ca_o and K_o represent the differences of corresponding ionic concentrations at certain time to the resting state. So the initial values of them are zero. Cap in Equation 5 denotes the capacity of cells to recover from the CSD and R in Equation 6 stands for the recovery mechanism, especially for relative refractory period.

(1) {\displaystyle \frac{\partial C{a}_a}{\partial t}}=D_{C{a}_a}{\nabla }^{\mathrm{2}}C{a}_a+v_a{K}_o-g_a(C{a}_a-C{a}_n)-f_a\tanh {\displaystyle \frac{C{a}_a}{C{a_a}^{\mathrm{0}}}}-k_{a}C{a_a}^{\mathrm{2}}-r_{a}RC{a}_a.

(2) {\displaystyle \frac{\partial C{a}_n}{\partial t}}=g_n(C{a}_a-C{a}_n)-f_n\tanh {\displaystyle \frac{C{a}_n}{C{a_n}^0}}-k_{n}C{a_n}^2-r_{n}RC{a}_n.

(3) C{a}_o=-{\displaystyle \frac{\gamma }{\alpha }}C{a}_a-{\displaystyle \frac{\beta }{\alpha }}C{a}_n.

(4) {\displaystyle \frac{\partial K_o}{\partial t}}=D_{K_o}{\nabla }^{\mathrm{2}}{K}_o+v_{n}C{a}_n-f_o\tanh {\displaystyle \frac{K_o}{{K_o}^{\mathrm{0}}}}-k_o{K_o}^{\mathrm{2}}-r_{o}R{K}_o.

(5) \{\begin{array}{c}{\displaystyle \frac{\partial Cap}{\partial t}}={\displaystyle \frac{\mathrm{1}-Cap}{T_{\mathrm{0}}}}, \mathrm{if} C{a}_a>ca \mathrm{or} C{a}_n>cn \mathrm{or} K_o>ck,\\ {\displaystyle \frac{\partial Cap}{\partial t}}=-{\displaystyle \frac{Cap}{T_f}}, \mathrm{if} C{a}_a\le ca \mathrm{and} C{a}_n\le cn \mathrm{and} K_o\le ck.\end{array}

Here ca, cn and ck are thresholds related to the recovery from CSD. Biological meanings and values of the parameters are listed in Table 2.

(6) \{\begin{array}{c}{\displaystyle \frac{\partial R}{\partial t}}={\displaystyle \frac{\mathrm{1}-R}{t_{\mathrm{0}}}}, \mathrm{if} Cap\mathrm{>ccap}\\ {\displaystyle \frac{\partial R}{\partial t}}=-{\displaystyle \frac{R}{t_f}}, \mathrm{if} Cap\le \mathrm{ccap}\end{array}

Here ccap is the threshold related to the opening of the recovery mechanism. We fixed part of values of the parameters in this model following the algorithm put forward by Chapuisat [ 40, 49].

Implementation

All the computations and visualizations of the model were implemented in the Matlab environment (Matlab7.0, the MathWorks Inc., USA). An explicit difference method was used to solve the differential equations [ 33, 51]. The computations were simulated in a one-dimensional region of the cortex [ 52]. The time and the space were divided into discrete steps with temporal precision of 1 s and spatial precision of 0.2 mm respectively. Since high extracellular potassium was considered as the driving force sustaining CSD propagation [ 53, 54], an injection of KCl with concentration 20 mM around 1 mm area was used to initiate CSD. For transient stimulation, the stimulation was only kept for 1 s. The continuous stimulation was kept for an assigned time (e.g., 3600 s) in this paper. The propagation dynamics of CSD waves was represented as the change of extracellular potassium concentration.

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