Optogenetic induced epileptiform activity in a model human cortex

We use the meso-scale cortical model developed by Liley et al. (2001) to simulate the dynamics of a human cortex. The non-dimensionalised form of this model can be found in the appendix.

where g _{C h R2} is the total conductance of the optogenetic channels in the O1 and O2 states. U ₀ and U ₁ are empirical constants and N _{C h R2} is the number of ChR2 channels per representative neuron. The expression for g _{C h R2}, and values and explanations for all parameters can be found in (Selvaraj et al. 2014).

We now present results obtained by optogenetic stimulation of a portion of model cortex of dimensions 1400×1400 mm², which is divided into 100×100 cells or representative neurons^a. We stimulate an area of model cortex with spatial scale of the order of clinical recordings. In Figure 1, a square region in the middle of the cortex of approximate dimensions 280×280 mm² (1/25 the total area of the cortex) is modified to express ChR2 ion channels in the excitatory population, and this region is illuminated with light of a constant 50 mW /mm² intensity starting at 0.5 s. An ion channel density of 10⁹ChR2^′s/m ² is used to keep illumination intensities within physiologically permissible values and still ensure optimal stimulation of the cortex.

Figure 1 depicts the birth of a seizure-like wave within the square patch of stimulated cortex. Neighbouring regions are then excited by the outward propagation of travelling seizure waves, and the effects of the stimulation spread to the entire cortical area. Hyper-excitation in (Selvaraj et al. 2014) was achieved by increasing subcortical inputs to the excitatory population throughout the model cortex. This meant the entire cortex was on the verge of seizing, and a small increase in subcortical inputs to a column could cause a region of cortex to seize. These seizures originated from a cortical column, and propagated outwards in spiral waves.

Here, spatially uniform travelling waves are formed by the equal hyper-excitation of cortical macrocolumns within a patch of cortex using optogenetic channels, and subcortical inputs do not play as important a role in starting seizures as illumination intensity increases.

It should be noted here that the entire model cortex is functioning normally, with inhibitory and excitatory inputs of comparable magnitudes balancing each other out. However, when adequate optogenetic stimulation is applied to the excitatory population, it depolarises these cells, increasing their mean soma potential, which immediately increases their firing rate. The local excitatory and long range contributions to post synaptic activation are increased because of the higher firing rate, producing a further rise in the mean soma potential of both populations, which ultimately leads to even higher firing rates in both excitatory and inhibitory cells. Increases in inhibitory firing rates trail excitatory ones within a macrocolumn by 1-2 ms for two reasons. One, stimulation is applied directly to the excitatory population, and two, there are time delays associated with synaptic transmission. Spatial connectivity between columns transfers synchronous activity to neighbouring columns of neurons outside the stimulated cortical patch, exciting them into synchronous states facilitating propagation of seizure activity. As stated earlier, only neurons within the stimulated area are hyperexcited while the rest of the cortex is functioning normally. This constrains the minimum stimulated area necessary to excite the rest of the cortex into an epileptic state to ∼4% of the total area of the cortex. Smaller stimulated regions are not able to support seizure activity for cortical and optogenetic parameters mentioned above.

In Figure 2, we look at the variation of mean soma potential of the excitatory population at a point within the stimulated patch of cortex, and take a one dimensional cross section of the two dimensional spatial domain to study how different illumination intensities and illumination profiles lead to travelling seizure waves of varying frequencies. For all three illumination profiles presented in Figure 2, stimulation is turned on at t = 0 s. Figures 2a and 2d show the variation of mean soma potential with time when the cortical patch is constantly illuminated with intensities of 30 mW /mm² and 60 mW /mm², respectively, while Figure 2g uses a 200 ms on 200 ms off pulsed illumination of 60 mW /mm² intensity as shown in Figure 2i. Figures 2b, 2e and 2h depict travelling waves in a one dimensional slice of the two dimensional domain, and comprises both stimulated cortex between 560 mm and 840 mm, and normally functioning cortex.

Higher illumination intensities result in higher conductances, as shown in Figures 2c and 2f, resulting in cells being depolarised more quickly. For a given illumination intensity, pulsed light can reduce the frequency of seizure waves because the stimulatory input rapidly drops to zero when light is turned off, sending the cortex back to a normally functioning state. In other words, increasing the time of no illumination when using a pulsed light source decreases seizure frequency. This difference can be seen between Figure 2d, where constant illumination is used, and Figure 2g, where pulsed illumination is used. Counter-intuitively, though, while a higher intensity depolarises cells more quickly, it reduces the frequency of seizure waves as seen by comparing Figures 2a and 2d. One possible explanation is that the rate of change of mean soma potential given in equation 1 is lower for higher intensities because the stimulation term, u, is always positive on account of using ChR2, a cation pump. A higher rate of change decreases the time required to change from a lower to a higher firing rate and back. In other words, the rate of change of firing rate increases, which results in higher frequency oscillations.

We turn to bifurcation analysis to give us a picture of what happens in the cortex when illumination of a certain intensity is used. To perform a bifurcation analysis, we turn off the stochastic and spatial terms in the cortical model to obtain an underlying set of deterministic ordinary differential equations (ODEs), which helps us gain insight into the complete system of stochastic partial differential equations (SPDEs) that describe the mesoscale cortical model intuitively. It has been shown that Hopf bifurcations in the dimensionless ODEs can correspond to travelling waves in the SPDE system (Kramer et al. 2007). Here, we combine this ODE system with the optogenetic model to study the dynamics of the combined system.

Figure 3a shows the values of Γ _e and P _ee that cause oscillatory behaviour in the ODE model when not stimulated (black) and when stimulated (grey) by optogenetic channels using an illumination intensity of 60 mW /mm². The red dot in the figure is located at Γ _e = 0.0012 and P _ee = 11.0, which are used in the full SPDE system simulations shown in Figures 1 and 2. These values lie well outside the region of epilepsy in the unstimulated cortex. The seizure prone area in the parameter space is vastly increased when optogenetic stimulation is applied. It has been observed, but not shown here, that this area is slightly larger with less distinct boundaries for the SPDE system owing to stochasticity.

A bifurcation analysis of the ODE system yields a bifurcation diagram depicting the salient features for the mean soma potential of the excitatory population, h _e , for different illumination intensities as shown in Figure 3b. We use P _ee = 11.0 and Γ _e = 0.0012 for this analysis. Solid lines indicate stable fixed points, while dashed lines represent unstable ones. Dot-dashed lines and dotted lines indicate maximum and minimum values of h _e achieved during stable and unstable limit cycles, respectively. The asterisk represents a subcritical Hopf bifurcation at 5.1352 mW /mm², which gives rise to an unstable limit cycle. At 4.7908 mW /mm² the limit cycle stabilises after going through a saddle node bifurcation, and becomes unstable again after going through another saddle node bifurcation at 33.8682 mW /mm². While Figure 3b is truncated at 40 mW /mm², the limit cycle remains unstable and does not terminate well beyond 100 mW /mm², which is around the limit of physiologically acceptable illumination intensities^b. This suggests the combined cortical-optogenetic model is unable to support stable oscillations past 34 mW /mm², and at higher intensities, we might be observing a succession of independent columnar spiking events and not entrained spiking seen during stable oscillatory behaviour. In the case of an individual spike in mean soma potential, a cortical column relaxes to its resting potential before firing again. This takes longer to produce a spike than the case with continuous oscillatory behaviour, where a column can be excited to spike again before reaching resting state. This explains why lower frequency seizure waves are observed when higher illumination intensities are used, a trend which is shown in Figure 3c after averaging frequency over multiple simulations of the full SPDE system. For lower intensities, subcortical inputs may aid in exciting or suppressing the system, so the probability of producing seizures decreases for intensities lower than 25 mW /mm². For higher intensities, the system will almost always produce seizures, but the frequency of seizure waves is dependent on the magnitude of subcortical and long range inputs. Overall, however, we observe a tendency to produce lower seizure wave frequencies for higher illumination intensities.

^a The average human cortex has dimensions of 500×500 mm² if it were laid open like a sheet. However, to remain consistent with previous work, and because dynamics in this model of undifferentiated cortex are scale-free, we have used a larger cortical domain to illustrate seizure waves.

^b Prolonged exposure (>0.5 s) at an intensity of 100 mW /mm² caused significant tissue damage in animal models (Cardin et al. 2010).

The non-dimensional meso-scale model was first stated in (Kramer et al. 2006), where values and explanations for all variables and constants of the model can be found. For convenience, we state the equations here.

All variables have been non dimensionalised and are functions of time \(\tilde {t}\), and the two spatial dimensions \(\tilde {x}\) and \(\tilde {y}\). The subscripts e and i represent excitatory and inhibitory populations respectively, and variables with two subscripts represent the transfer of energy from one population to another. The mean soma potential for a neuronal population is represented by the \(\tilde {h}\) state variable, \(\tilde {I}\) represents the postsynaptic activation due to local, long-range, and subcortical inputs. \(\tilde {\phi }\) represents long range (corticocortical) inputs.

where N _Oi and N _Ci represent the fraction of channels in the open and closed states, respectively. K _ai are the rates of transition from the closed states, C1 and C2, to the open states O1 and O2 respectively. Conversely, K _di are the closing rates from the open states to the closed states. K _r is the thermal recovery rate from C2 to C1. e ₁₂ and e ₂₁ are the transition rates from O1 to O2 and vice versa. The values for all rate constants can be found in table (Selvaraj et al. 2014).