Outgrowing seizures in Childhood Absence Epilepsy: time delays and bistability

Abstract

We formulate a conductance-based model for a 3-neuron motif associated with Childhood Absence Epilepsy (CAE). The motif consists of neurons from the thalamic relay (TC) and reticular nuclei (RT) and the cortex (CT). We focus on a genetic defect common to the mouse homolog of CAE which is associated with loss of GABA_A receptors on the TC neuron, and the fact that myelination of axons as children age can increase the conduction velocity between neurons. We show the combination of low GABA_A mediated inhibition of TC neurons and the long corticothalamic loop delay gives rise to a variety of complex dynamics in the motif, including bistability. This bistability disappears as the corticothalamic conduction delay shortens even though GABA_A activity remains impaired. Thus the combination of deficient GABA_A activity and changing axonal myelination in the corticothalamic loop may be sufficient to account for the clinical course of CAE.

Appendix: Intrinsic currents

We present here the details of the models for the intrinsic currents of the three neurons of the model (1).

Leak currents:

TC neurons (Destexhe et al. 1993, 1998): $I_{\mathrm {L}} = \bar {g}_{\mathrm {L}}(V-E_{\mathrm {L}})$; $\bar {g}_{\mathrm {L}}$= 0.01 mS/cm² and E_L = -70 mV.

CT neurons (Destexhe et al. 1998): $I_{\mathrm {L}} = \bar {g}_{\mathrm {L}}(V-E_{\mathrm {L}})$; $\bar {g}_{\mathrm {L}}$ = 0.1 mS/cm² and E_L = -70 mV.

RT neurons(Destexhe et al. 1996): $I_{\mathrm {L}} = \bar {g}_{\mathrm {L}}(V-E_{\mathrm {L}})$; $\bar {g}_{\mathrm {L}}$ = 0.05 mS/cm² and E_L = -90 mV.

Transient voltage-gated K⁺ current (Traub and Miles 1991)

$$\begin{array}{@{}rcl@{}} I_{\mathrm{K}} &=& \bar{g}_{\mathrm{K}}{m_{k}^{4}} h_{\mathrm{K}} (V-E_{\mathrm{K}}) \end{array} $$

(12)

$$\begin{array}{@{}rcl@{}} \frac{dm}{dt} &=& \alpha_{m}(V)(1-m) - \beta_{m}(V)m \end{array} $$

(13)

$$\begin{array}{@{}rcl@{}} \frac{dh}{dt} &=& \alpha_{h}(V)(1-h) - \beta_{h}(V)h \end{array} $$

(14)

$$\begin{array}{@{}rcl@{}} \alpha_{m_{k}}(V) &=& \frac{0.032(15-V)}{\exp(\frac{15-V}{5})-1} \end{array} $$

(15)

$$\begin{array}{@{}rcl@{}} \beta_{m_{k}}(V) &=& 0.5 \exp(\frac{10-V}{40}) \end{array} $$

(16)

$$\begin{array}{@{}rcl@{}} \alpha_{h_{k}}(V) &=& 0.028\exp(\frac{15-V}{15})+\frac{2}{\exp(\frac{85-V}{10})+ 1} \end{array} $$

(17)

$$\begin{array}{@{}rcl@{}} \beta_{h_{k}}(V) &=& \frac{0.4}{\exp(\frac{40-V}{10})+ 1} \end{array} $$

(18)

TC neurons: $\bar {g}_{\mathrm {K}}=$ 10 mS/cm², E_K = -90 mV

CT neurons: $\bar {g}_{\mathrm {K}}=$ 5 mS/cm², E_K = -90 mV

RT neurons: $\bar {g}_{\mathrm {K}}=$ 20 mS/cm², E_K = -80 mV

Transient voltage-gated Na⁺ current (Traub and Miles 1991)

$$\begin{array}{@{}rcl@{}} I_{\text{Na}} &=& \bar{g}_{\text{Na}}m_{\text{Na}}^{3} h_{\text{Na}} (V-E_{\text{Na}}) \end{array} $$

(19)

$$\begin{array}{@{}rcl@{}} \frac{dm}{dt} &=& \alpha_{m}(V)(1-m) - \beta_{m}(V)m \end{array} $$

(20)

$$\begin{array}{@{}rcl@{}} \frac{dh}{dt} &=& \alpha_{h}(V)(1-h) - \beta_{h}(V)h \end{array} $$

(21)

$$\begin{array}{@{}rcl@{}} \alpha_{m_{\text{Na}}}(V) &=& \frac{0.32(13-V)}{\exp(\frac{13-V}{4})-1} \end{array} $$

(22)

$$\begin{array}{@{}rcl@{}} \beta_{m_{\text{Na}}}(V) &=& \frac{0.28(V-40)}{\exp(\frac{V-40}{5})-1} \end{array} $$

(23)

$$\begin{array}{@{}rcl@{}} \alpha_{h_{\text{Na}}}(V) &=& 0.128 \exp(\frac{17-V}{18}) \end{array} $$

(24)

$$\begin{array}{@{}rcl@{}} \beta_{h_{\text{Na}}}(V) &=& \frac{4}{\exp(\frac{40-V}{5})+ 1} \end{array} $$

(25)

for TC neurons: $\bar {g}_{\text {Na}}$ = 90 mS/cm², E_Na = + 45 mV

for CT neurons: $\bar {g}_{\text {Na}}$ = 50 mS/cm², E_Na = + 45 mV

for RT neurons: $\bar {g}_{\text {Na}}$ = 200 mS/cm², E_Na = + 45 mV

Low threshold Ca⁺⁺ current (Huguenard and Prince 1992)

$$\begin{array}{@{}rcl@{}} I_{TS} &=& \bar{g}_{\text{Ca}} m_{\text{TS}}^{2} h_{\text{TS}} (V-E_{\text{Ca}}) \end{array} $$

(26)

$$\begin{array}{@{}rcl@{}} \frac{dm}{dt} &=& \frac{m_{\infty}(V) -m}{\tau_{m}(V)} \end{array} $$

(27)

$$\begin{array}{@{}rcl@{}} \frac{dh}{dt} &=& \frac{h_{\infty}(V) -h}{\tau_{h}(V)} \end{array} $$

(28)

$$\begin{array}{@{}rcl@{}} m_{\infty}(V)&=& \frac{1}{1+\exp(\frac{-(V + 52)}{7.4})} \end{array} $$

(29)

$$\begin{array}{@{}rcl@{}} \tau_{m}(V)&=& 1+\frac{0.33}{\exp(\frac{V + 48}{4})+\exp(\frac{-(V + 407)}{50})} \end{array} $$

(30)

$$\begin{array}{@{}rcl@{}} h_{\infty}(V)&=& \frac{1}{1+\exp(\frac{V + 80}{5})} \end{array} $$

(31)

$$\begin{array}{@{}rcl@{}} \tau_{h}(V)&=& 28.3 + \frac{0.33}{\exp(\frac{V + 48}{4})+\exp(\frac{-(V + 407)}{50})} \end{array} $$

(32)

where $\bar {g}_{\text {Ca}}= 3$ mS/cm², E_Ca = + 120 mV.

Depolarization-activated K⁺ current (McCormick et al. 1993)

$$\begin{array}{@{}rcl@{}} I_{\mathrm{M}} &=& \bar{g}_{\mathrm{M}} m_{\mathrm{M}} (V-E_{\mathrm{M}}) \end{array} $$

(33)

$$\begin{array}{@{}rcl@{}} \frac{dm}{dt} &=& \frac{m_{\infty}(V) -m}{\tau_{m}(V)} \end{array} $$

(34)

$$\begin{array}{@{}rcl@{}} m_{\infty}(V) &=& \frac{1}{1+\exp(\frac{-(V + 35)}{10})} \end{array} $$

(35)

$$\begin{array}{@{}rcl@{}} \tau_{m}(V) &=& \frac{1000}{3.3\exp(\frac{V + 35}{20}) +\exp(\frac{-(V + 35)}{20})} \end{array} $$

(36)

where $\bar {g}_{\mathrm {M}}= 0.07$ mS/cm², E_M = − 100 mV.

Low-threshold Ca⁺⁺, depolarization-activated hyperpolarization and slow K⁺ currents (Destexhe et al. 1993; Destexhe and Babloyantz 1991; Huguenard and Prince 1991; Wang et al. 1991)

$$\begin{array}{@{}rcl@{}} I_{T} &=& -\bar{g}_{\mathrm{T}}m_{\mathrm{T}}^{3} h_{\mathrm{T}} (V-E_{\mathrm{T}}) \end{array} $$

(37)

$$\begin{array}{@{}rcl@{}} I_{h} &=& \bar{g}_{h} S F (V-E_{h}) \end{array} $$

(38)

$$\begin{array}{@{}rcl@{}} I_{K2} &=& \bar{g}_{\text{K2}} m_{\text{K2}}(0.6h_{\text{K2,1}} + 0.4 h_{\text{K2,2}})(V-E_{\mathrm{K}}) \end{array} $$

(39)

where $\bar {g}_{\mathrm {T}}=$2 mS/cm², g_h =0.02 mS/cm², g_K2 =0.00005 mS/cm², E_T =+ 120 mV, E_h =-43 mV, E_K2 = -90 mV. The dynamics for the gating variables m_T, h_T, S, F, m_K2, h_K2,1, h_K2,2 have non-standard forms, and can be found in Table 1 of Destexhe et al. (1993).