Mathematical Model of a Main Rhythm in Limbic Seizures

Abstract

While synchronization in the brain neural networks has been studied, the emergency of the main oscillation frequency and its evolution at different normal and pathological states remains poorly investigated. We propose a new concept of the formation of a main frequency in limbic epilepsy. The idea is that the main frequency is not a result of the activity of a single cell, but is formed due to collective dynamics in a ring of model neurons connected with delay. The individual cells are in an excitable mode providing no self-oscillations without coupling. We considered the ring of a different number of Hodgkin–Huxley neurons connected with synapses with time delay. We have shown that the proposed circuit can generate oscillatory activity with frequencies close to those experimentally observed. The frequency can be varied by changing the number of model neurons, time delay in synapses and coupling strength. The linear dependence of the oscillation period on both coupling delay and the number of neurons in the ring was hypothesized and checked by means of fitting the values obtained from the numerical experiments to the empirical formula for a constant value of coupling coefficient.

1. Introduction

Synchronization in neural networks traditionally attracts great interest when studying the brain. In particular, for epilepsy research and modelling, a mathematical description of synchronization is very important [1,2], since epilepsy is characterized by the patterns of highly synchronous activity [3]. However, studying only the synchronization is not enough to understand the nature of epilepsy. Limbic epilepsy (also known as temporal lobe epilepsy [4]) is the most common type of epilepsy in humans [5,6]. In secondary generalized forms, such as limbic epilepsy, epileptic rhythms including the main frequency of epileptiform activity in the brain occurs in very small groups of neurons [7]. Only then does the pathological activity spread to large brain networks.

For primary generalized epilepsies, there is a long history of modelling. In some early papers [8,9] epileptic regime was considered to be a result of bifurcation. Then, such ideas became less popular due to them not explaining the observed phenomena well. In particular, bifurcation mechanisms should mostly occur due to changes in ion conductivity [10] or due to significant changes in the connectivity architecture of cells [11]. But for many forms of epilepsy, including limbic epilepsy in rodents, there could be tens of seizures per hour [12]. If we accept a “bifurcation theory of epilepsy” this means that there are reversible chemical reactions or reversible anatomical changes with the time scale of some seconds to tens of seconds which are repeated 100 or more times during 8 h of recording. This seems to be very unlikely. Instead, the ideas of coexisting attractors have became more popular [13,14]: low-amplitude (possibly noise-induced) oscillations with a wide basin (this regime corresponds to a normal behaviour) and high-amplitude oscillations with a narrow basin (this is the pathological regime). Transition between these regimes can occur by means of either noise or external input (e.g., sensory stimulus from trigeminal nerve). Some recent papers also provide a third idea that the epileptic state may be a long-lasting transient which can occur due to a specific organization of the brain network near the limit cycle saddle-node bifurcation [15,16].

Little has been conducted till now for modelling the generator of the principal rhythm of limbic epileptic [17,18]. Considering the larger problem, i.e., the generation of main frequencies of different brain rhythms, the recent paper [19] is of a great interest. In general, there could be two approaches to the construction of mathematical models of brain rhythms. The first approach is to construct a network based on some physiological ideas, and then to investigate it in order to find parameter values at which the network exhibits the desired behaviour. Previously, such an approach was used for modelling thalamocortical system [15,20,21]. The other approach is to construct the desirable behaviour manually in the phase space using well-known subsystems as building blocks as it was performed in [22,23]. This second approach is used here.

In this work, we construct a dynamical system for which the oscillatory regime of interest is directly specified by its properties (assumed by construction). The system is a ring of pyramidal neurons of the hippocampus connected unidirectionally. For individual neurons, the Hodgkin–Huxley equations [24] are written with parameters taken from physiological considerations [25]. When an pulse is generated by a single neuron, it is transmitted to the next in the ring. If the ring is long enough for the pulse to return after the first neuron has left the refractory state, the first neuron fires again and thus periodic oscillations occur. The possibility of the existence of such an oscillatory regime should be determined by the coupling strength and the time of signal propagation along the ring. Proving the existence and investigation of this regime is the primary task of this work.

2. Model

The main model in this work is a ring of identical Hodgkin–Huxley neurons [24] unidirectionally connected with a delay. An example of the coupling architecture for a system of 11 elements is shown in Figure 1. The neurons No. 1–10 make up the main oscillatory ring. Neuron No. 11 was used as an external driving force to begin oscillations; with its help, the system was transferred from the stable point to the oscillatory mode. We also considered the option when, instead of a single neuron, a chain of elements was used to switch to the oscillatory mode; but such a technique did not provide any significant advantages. The equation for each neuron can be written as follows.

$$\begin{array}{cc}\hfill \frac{d{V}_i}{dt}& =[I_{stim}-g_{Na}{m}_i^3{h}_i(V_i-V_{Na})-g_K{n}^4(V_i-V_K)-g_L(V_i-V_L)+{\kappa }_i]/C,\hfill \\ \hfill \frac{d{m}_i}{dt}& ={\alpha }_m\left(V_i\right)(1-m_i)-{\beta }_m\left(V_i\right)m_i,\hfill \\ \hfill \frac{d{n}_i}{dt}& ={\alpha }_n\left(V_i\right)(1-n_i)-{\beta }_n\left(V_i\right)n_i,\hfill \\ \hfill \frac{d{h}_i}{dt}& ={\alpha }_h\left(V_i\right)(1-h_i)-{\beta }_h\left(V_i\right)h_i,\hfill \end{array}$$

(1)

where V is the membrane potential measured across the membrane of the neuron; C is the membrane capacitance; $g_{Na}$, $g_K$ and $g_L$ are the maximum conductivities of sodium and potassium channels and leakage, respectively; $V_{Na}$ and $V_K$ are reciprocal potentials for sodium and potassium ions; and m, n and h are the gate characteristics of various types of ion channels: m and n describe the activation of sodium and potassium channels, respectively, and h denotes the deactivation of the sodium channel.

In the numerical experiment, we mainly used the parameters of neurons from [25]: C = 1 mF/cm${}^2$, $g_{Na}$ = 40 mS/cm${}^2$, $g_K$ = 35 mS/cm${}^2$, $g_L$ = 0.3mS/cm${}^2$, $V_{Na}$ = 55 mV, $V_K$ = −77 mV, and $V_L$ = −65 mV. These parameters make it possible to obtain dynamics corresponding to self-oscillations in rodent pyramidal neurons. Since we were not interested in self-oscillations in a single neuron, except for one neuron used for network excitation (in Figure 1 this is neuron No. 11), we changed the value of $V_L$ = −66.8 mV for all other neurons in the ring to obtain a sub-threshold excitable regime. A special neuron was in oscillatory mode for a short time, then it was disabled (it’s oscillations is shown in Figure 2a)

The functions ${\alpha }_i$ and ${\beta }_p$ ($p\in \{n,m,h\}$) were taken from the same work [25] to reproduce the dynamics of an axon of a pyramid cell.

$$\begin{array}{cccc}\hfill {\alpha }_n\left(V\right)& =\frac{0.8(V-25)}{1-e^{-(V-25)/9}},\hfill & \hfill {\beta }_n\left(V\right)& =\frac{-0.002(V-25)}{1-e^{(V-25)/9}},\hfill \\ \hfill {\alpha }_m\left(V\right)& =\frac{0.182(V+35)}{1-e^{-(V+35)/9}},\hfill & \hfill {\beta }_m\left(V\right)& =\frac{-0.124(V+35)}{1-e^{(V+35)/9}},\hfill \\ \hfill {\alpha }_h\left(V\right)& =0.25e^{-(V+90)/12},\hfill & \hfill {\beta }_h\left(V\right)& =\frac{0.25e^{-(V+62)/6}}{e^{(V+90)/12}}.\hfill \end{array}$$

(2)

The total effect of the system on each i-th neuron was set using the coupling function ${\kappa }_i$ as follows:

$${\kappa }_i\left(t\right)=\sum _{j=1}^D{k}_{i,j}(1+tanh\left(V_j(t-\tau )\right)),$$

(3)

where D is the total number of neurons in the system, $k_{i,j}$ is the coefficient of impact of the j neuron on i, and $\tau$ is the time delay parameter. The connection was introduced in such a way that, at high-voltage values, the neuron ceased to respond to the driving. All coupling coefficients were set to either $k_{i,j}=k$ for enabled connections or $k_{i,j}=0$ for disabled ones.

The network was studied in a bistable regime. The first attractor was a stable fixed point. This regime was inherited from the phase space of individual neurons. The second attractor was a limit cycle, it was of particular interest for us. This cycle appeared due to coupling. It was theoretically hypothesized based on properties of the Hodgkin–Huxley model as described in the Introduction. Since the oscillatory regime emerges with the finite amplitude at some value of k and coexists with the fixed point, the only way it can be born is via a saddle-node bifurcation of a limit cycle [26,27]. In the parameter space, both $V_L$ and k can be bifurcation parameters.

3. Simulation

We performed simulations for four different values of k: 30, 40, 50 and 60. For each k value we examined the system for 0.2 ms $⩽\tau ⩽$ 1 ms changes with a step of 0.05 ms (let us denote them ${\tau }_i$, $i=1,\dots ,i_{max}$, $i_{max}=17$) and $10⩽D⩽100$ changes by a step of 5 (let us denote them $D_j$, $j=1,\dots ,j_{max}$, $j_{max}=19$). Since the considered systems has a time delay, it was numerically integrated using second-order Runge–Kutta method with time step $\Delta$t = 0.01 ms.

The network demonstrated two regimes. The first one was a fixed point which was inherited from uncoupled oscillators due to the synaptic coupling (3) tending towards zero for $V_j<<0$. The second regime was periodic oscillations. This regime was born due to saddle-node bifurcation of a limit cycle (not due to Andronov–Hopf bifurcation); therefore, it coexists with a fixed point. The bifurcation parameters are D, $\tau$ and k and an increase in any of them for some reasonable values of others leads to the appearance of the oscillatory regime. Due to bistability in the system, to reach the oscillatory regime we used limited (and short) time external driving from the external neuron No. $(D+1)$. The end time moment of this driving is shown by a blue line in Figure 2 (the starting time of the driving is $t=0$).

The studied network was constructed to produce periodical oscillations, modelling the main rhythm of epileptic activity. Therefore, the simulation mostly aimed to prove the existence of the oscillatory regime and to obtain the dependence of the main oscillation frequency f on the number of neurons D and the coupling delay $\tau$. So, we studied the dynamics at those values of k, D and $\tau$ at which the oscillatory regime appears, see Figure 3 where f is plotted by colour.

Considering Figure 3, one can suggest that the dependency of f on D and $\tau$ is hyperbolic. Let us formulate this hypothesis and test it.

4. Approximation

It is easier to consider linear dependency rather than hyperbolic dependency. Therefore, let us formulate our hypothesis about the period of oscillations T rather than about the frequency f, assuming $f=1/T$.

Hypothesis 1. 

For the fixed coupling coefficient k the oscillation period in the ring of unidirectionally coupled Hodgkin–Huxley neurons (1) depends only on the number D of oscillators in the ring and the delay time τ, with this dependence being the linear superposition of D and $\tau D$ with constant parameters.

To prove or disprove the formulated hypothesis analytically, it is necessary to obtain an exact solution for the Equation (1). For much simpler systems of phase oscillators similar problem was studied analytically in [28]. But for a ring of Hodgkin–Huxley neurons this does not seem achievable in the current state of science. Therefore, we chose another way. We solved the system (1) numerically for different D and $\tau$ values and calculated the oscillation period T in each case. Then, we fitted the dependency $T(D,\tau )$ reported above to the following Formula (4):

$$\tilde{T}(D,\tau ,k)=T_0\left(k\right)+\gamma \left(k\right)\tau D+\epsilon \left(k\right)D,$$

(4)

where $\tilde{T}$ is the approximation of the period T measured in the simulation, and $T_0$, $\epsilon$ and $\gamma$ are constants depending only on k, with $T_0$ and $\epsilon$ possessing the dimension of time and $\gamma$ being dimensionless. We should note that $\epsilon$ is a parameter of inertia of a single neuron with an axon (as in most time delay systems, see [29,30]), i.e., the time shift which the neuron provides itself without coupling delay $\tau$. The formula was fitted for each k separately using all $Q=i_{max}{j}_{max}$ values of T measured for different D and $\tau$ values. To estimate the goodness of fit, we calculated the normalized error of approximation ${\sigma }_{\mathrm{appr}}^2$ as follows:

$${\sigma }_{\mathrm{appr}}^2\left(k\right)=\frac{1}{Q{\sigma }_T^2\left(k\right)}\sum _{i=1}^{i_{max}}\sum _{j=1}^{j_{max}}{\left(T\left(k\right)-\tilde{T}\left(k\right)\right)}^2,$$

(5)

where ${\sigma }_T\left(k\right)$ is the empirical standard deviation for T at the fixed k.

Values of both reconstructed $\tilde{T}$ and measured T periods of oscillations are represented in Figure 4. These plots show that the reconstructed values are much closer to the measured ones with a greater number of neurons in the ring and with higher values of the coupling coefficient. The results of approximation are briefly summarized in the

Table 1. Values of the parameters $T_0$, $\epsilon$ and $\gamma$ in the Formula (4) fitted for different k with the mean-squared-normalized error of approximation ${\sigma }_{\mathrm{appr}}^2$.

k	${\mathit{T}}_0$, ms	$\mathit{\gamma }$	$\mathit{\epsilon }$, ms	${\mathit{\sigma }}_{\mathbf{appr}}^2$
30	14.27	0.985	3.08	$9.8\times {10}^{-4}$
40	13.49	0.960	2.00	$6.4\times {10}^{-4}$
50	10.81	0.960	1.49	$4.0\times {10}^{-4}$
60	8.79	0.950	1.21	$2.1\times {10}^{-4}$

. One can see that the approximation error is quite moderate and in all considered cases it is less than ${10}^{-3}$. This error can be expressed by both model imperfection and calculation errors. First, the period T was measured in simulation with a precision of up to $\Delta t=0.01$ ms, i.e., the approximated value was measured with a relative error ∼${10}^{-4}$. Second, the method of numerical integration was imperfect providing similar error $O(\Delta t)$. The approximation procedure also has some impact on the error, though the use of the LAPACK procedure from Python code (see [31]) should provide much less error than ∼${10}^{-4}$.

5. Conclusions and Discussion

In this study, a system of Hodgkin–Huxley neuron oscillators connected in a ring was considered. The coupling was unidirectional. The oscillations were excited using the short-term activity of a starting neuron. The non-zero coupling coefficients between neurons were equal. It was shown to be possible to control the signal frequency f by changing the number of neurons in the ring D and the coupling delay time $\tau$. Changing D affects the frequency change more strongly than $\tau$. In addition, increasing the values of the coupling coefficient between neurons in the ring made it possible to obtain modes with higher oscillation frequencies. The whole range of frequencies characteristic to limbic seizures [12] can be reproduced by the proposed approach.

Then, the dependence of T (the oscillation period) on D and $\tau$ was analysed. From this we formulated the hypothesis that this dependence is linear. To check this hypothesis, a mathematical model was constructed, the coefficients of which were fitted to the numerically obtained solutions. This fitting showed that the model has sufficiently high accuracy (relative error ${\sigma }_{\mathrm{appr}}^2<{10}^{-3}$).

We did not formulate our hypothesis as a theorem since we were not completely sure whether the Formula (4) was exact and an analytical proof or refutation is difficulty to perform. Increasing the calculation precision could provide some additional information if the approximation error reduces with a decrease in $\Delta t$. However, for modelling purposes the obtained precision is enough.

When fitting the Formula (4) to the simulated data, the three parameters behave very differently. The parameter $\gamma$ is mostly constant for all values of k, which seems to be natural. This parameter shows that the impact of the delay in the ring as it should not depend on the coupling from its physical sense. The parameter $\epsilon$ has a physical sense of an inertia parameter of an individual neuron. One can see that it is reduced with an increase in k. This means that unlike simple time delay systems [29], the Hodgkin–Huxley neuron the inertia is non-linear and the inertia coefficient depends on the signal amplitude and shape. In particular, a decrease in $\epsilon$ with an increase in k should be the result of more narrow impulses in the group for larger k values. It is also interesting that we obtained a non-zero value of $T_0$ in the model (4), which decreases with an increase in the coupling coefficient k. This constant cannot be described physically, since it gives a non-zero period for $D=0$. However, for small values of D the oscillatory regime, which was of particular interest to us, does not appear at all since the neurons are deeply depolarized and do not respond to the external impulse of the previous neuron.

In our simulations we observed the only oscillatory mode for each given set of parameters, i.e., there were only two stable regimes in the system: the stable point and the observed regime. However, one could theoretically expect that with a sufficiently large number of oscillators in the system, not one, but several oscillatory modes with multiple frequencies could exist. In addition to the main regime observed in the framework of this work, there could be regimes with two, three times, etc. lower frequencies. These modes corresponded to two, three, etc. simultaneously moving pulses along the ring.