The Epileptor Model: A Systematic Mathematical Analysis Linked to the Dynamics of Seizures, Refractory Status Epilepticus, and Depolarization Block

A, B, Equilibrium points of the Epileptor model with respect to x₀ for m  =  0.5 (A) and m  =  0 (B). z is the third coordinate of the vector equilibrium points. Stable nodes and saddles are labeled as blue plus sign markers and black dots, respectively. Stable and unstable foci are labeled as red squares-dotted and green squares, respectively. C–E, Time series (stochastic) of the Epileptor model exhibit a normal activity ( C), a nonoscillatory state ( D), and a periodic solution ( E). The parameters m and x₀ are: m = 0 and x₀ = −2.5 ( C), m  =  0 and x₀ = −0.9 ( D), and m = 0.5 and x₀ = −0.9 ( E).

Parameter space for equilibrium points. A, B, There are 9 regions in I_ext2 = 0.45 (A) and 7 regions in I_ext2 = 0 (B). The description of both parameter spaces is found in the Results section (Epileptor dynamics, Parameter space of equilibrium points).

The Epileptor model bifurcation diagram with respect to the slow variable z (m  =  0.5, I_ext2 = 0.45). The Z-lower (solid), Z-middle (dash-dotted), and Z-upper (dashed) branches consist of stable nodes, saddles, and unstable foci, respectively. Decreasing z, the Z-lower and Z-middle branches collide in an SN₁ bifurcation. Above the Z-curve, lower (dash-dotted) branch consists of saddles, and upper branch is divided into two sub-branches: one sub-branch (dashed) consists of unstable foci and another (dash-dotted) of saddles. Increasing z, the two sub-branches collide in an SN₂ bifurcation. The inset is their enlarged view. Decreasing z, upper (dashed) and lower branches above the Z-curve collide in an SN₄ bifurcation. Increasing z, the Z-upper branch and lower branch above collide in an SN₃ bifurcation. The -curve is the average value of x₁ for each z constant. Let x₀ = −1.6, the z-nullcline ( ) is at the Z-middle branch. A SLE occurs with a fold/homoclinic bifurcation. A saddle (S) periodic orbit separates the SLE attractor (right) and a stable periodic orbit LC (to the left, final orbit not shown). Deterministic trajectories are plotted on both sides of the separatrix S defining two basins of attraction (indicated by arrows). r  =  0.003 for LC and r  =  0.0007 for SLEs.

The Epileptor model bifurcation diagram with respect to the slow variable z (m  =  0, ). The Z-lower (solid), Z-middle (dash-dotted), and Z-upper (dashed) branches consist of stable nodes, saddles, and unstable foci, respectively. Decreasing z, the Z-lower and Z-middle branches collide in an SN₁ bifurcation. Above the Z-curve, lower (dash-dotted) branch consists of saddles, and the upper branch is divided into two sub-branches: one (solid) consists of stable foci and another (dash-dotted) of saddles. Increasing z, the two sub-branches collide in an SN₂ bifurcation. The inset is their enlarged view. Decreasing z, upper (dashed) and lower branches above the Z-curve collide in a SNIC bifurcation. Increasing z, the Z-upper branch and lower branch above collide in an SN₃ bifurcation. The -curve is the average value of x₁ for each z constant. Let x₀ = −1.6, the z-nullcline ( ) is at the Z-middle branch. A SLE occurs with a fold/circle bifurcation. A saddle (S) periodic orbit separates the SLE attractor (right) and a stable periodic orbit LC (to the left, final orbit not shown). Deterministic trajectories are plotted on both sides of the separatrix S defining two basins of attraction (indicated by arrows). For the right trajectory, r  =  0.0006, I.C = [0 −5 2.5 0 0 0.01] and T_s = [0 1337]. For the left trajectory, r  =  0.001, I.C = [0 −5 1 0 0 0.01], and T_s = [0 1000].

The Epileptor model bifurcation diagram with respect to the slow variable z (m  =  0, I_ext2 = 0). The Z-lower (solid), Z-middle (dash-dotted), and Z-upper (dashed) branches consist of stable nodes, saddles, and unstable foci, respectively. Decreasing z, the Z-lower and Z-middle branches collide in an SN₁ bifurcation. Increasing z, the Z-upper and Z-middle branches collide in an SN₂ bifurcation. Below the Z-upper branch, lower (dashed) and upper (dash-dotted) branches corresponding to unstable foci and saddles, respectively, collide in an SN₃ bifurcation. The Z-upper branch and the upper (dash-dotted) branch below collide as z decreases in an SN₄ bifurcation. The -curve is the average value of x₁ for each z constant. Let , the z-nullcline ( ) is at the Z-middle branch. A SLE occurs with a fold/homoclinic bifurcation. For the (deterministic) trajectories, r  =  0.0005, I.C = [0 −5 2.5 0 0 0.01] and T_s = [0:0.001: 1540].

The Epileptor model bifurcation diagram with respect to the slow variable z (m = −0.5, I_ext2 = 0). The Z-lower (solid) and Z-middle (dash-dotted) branches consist of stable nodes and saddles respectively. The Z-upper branch is divided into sub-branches separated by a Hopf bifurcation, H: one sub-branch (solid) consists of stable foci and another (dashed) consists of unstable foci. Decreasing z, the Z-lower and Z-middle branches collide in an SN₁ bifurcation. Increasing z, the Z-upper (solid) and Z-middle branches collide in an SN₂ bifurcation. Below the Z-upper branch, lower (dashed) and upper (dash-dotted) branches corresponding to unstable foci and saddles, respectively, collide in an SN₃ bifurcation. The Z-upper (dashed) branch and the upper (dash-dotted) branch below collide as z decreases in an SN₄ bifurcation. Let , the z-nullcline ( ) is at the Z-middle branch. A SLE occurs with a fold/Hopf bifurcation. For the (deterministic) trajectories, r  =  0.0007, I.C = [0 −5 2.65 0 0 0.01] and T_s = [0:0.001: 1028].

The Epileptor model bifurcation diagram with respect to the slow variable z (m = − 1, I_ext2 = 0). The Z-lower (solid), Z-middle (dash-dotted), and Z-upper (solid) branches consist of stable nodes, saddles, and stable foci, respectively. Decreasing z, the Z-lower and Z-middle branches collide in an SN₁ bifurcation. Increasing z, the Z-upper and Z-middle branches collide in an SN₂ bifurcation. Below the Z-upper branch, lower (dashed) and upper (dash-dotted) branches corresponding to unstable foci and saddles respectively, collide in an SN₃ bifurcation. The Z-upper (solid) branch and the upper (dash-dotted) branch below collide as z decreases in a SNIC bifurcation. Let x₀ = −1.6, the z-nullcline ( ) is at the Z-middle branch. A SLE reduces to a periodic switch between a nonoscillatory state and a NS, which occurs through a fold/fold bifurcation. For the (deterministic) trajectories, r  =  0.0008, I.C = [0.5 −5 2 0 0 0.01] and T_s = [0:0.001: 1400].

On the LC existence and dynamics. The Epileptor model bifurcation diagram (z, x₁) with respect to z (m  =  0.5, I_ext2 = 0.45, x₀ = −1.6). A, B, The z-nullcline is given by Equation 34 ( A), and results from Equation 35 ( B). The -curve is the average value of x₁ for each z constant. C, D, Characteristic time series (deterministic) show a divergence of the Epileptor with time ( C), and the fast-slow limit cycle LC ( D). r  =  0.009 for LC (i₂) and r  =  0.0007 for SLEs (i₁).

On the LC existence and dynamics. The Epileptor model bifurcation diagram (z, x₁) with respect to z as plotted in Figure 4. The -curve is the average value of x₁ for each z constant. Let x₀ = −0.9, the z-nullcline is at three branches of the (z, x₁) curve. Two branches (dashed) consist of unstable foci and one branch (dash-dotted) consists of saddles (see inset). Deterministic trajectory converges to LC. The final state is the intersection of the z-nullcline and -curve. m  =  0.5, I_ext2 = 0.45, r  =  0.0035, I.C = [0 −5 3 0 0 0.01] and T_s = [0 1000].

A, B, Finding periodic orbits of the Epileptor for m  =  0.5 ( A) and m = −16 ( B). Stable periodic orbits LC and SLC are labeled as red plus sign markers (bottom) and blue plus sign markers (top), respectively. Saddle periodic orbits S are labeled as black dots. A, B, Decreasing x₀, periodic orbits LC and S disappear through a saddle-node of periodic orbits bifurcation. B, Decreasing further x₀ (below −0.8), SLC disappears.

Parameter space of periodic orbits. A, B, There are five regions in I_ext2 = 0.45 ( A) and I_ext2 = 0 ( B). LC exists in area I and coexists with S in area II. Only SLC exists in area IV, and coexists with LC and S in area III. LC, SLC, and S disappear in area V. We can visualize in Figure 5 the coexistence of LC and S (A, area II), and in Figure 13 the coexistence of LC, S, and SLC (B, area III).

SLC dynamics. A, B, The equilibrium point is a saddle (m  =  1, x₀ = −1.8; A) and an unstable focus (m = −2, x₀ = 0; B) with I_ext2 = 0, r  =  0.0007. C, D, Deterministic time series of the Epileptor system ψ, subsystem 1 ψ₁ and subsystem 2 ψ₂ are plotted for the saddle equilibrium point ( C) and for the unstable focus ( D). The -curve is the average value of x₁ for each z constant.

Transition from epileptiform fast discharges to depolarization block. A, Bifurcation diagram (z, x₁) of the Epileptor model. The Z-lower (solid), Z-middle (dash-dotted), and Z-upper (solid) branches consist of stable nodes, saddles, and stable nodes, respectively. Decreasing z, the Z-lower and Z-middle branches collide in an SN₁ bifurcation. Increasing z, the Z-upper and Z-middle branches collide in an SN₂ bifurcation. Below the Z-upper branch (see inset), the top (dash-dotted) branch consists of saddles and the bottom (dashed) branch consists of unstable foci. Increasing z, bottom and top branches collide in an SN₃ bifurcation. Decreasing z, the Z-upper (solid) branch and upper (dash-dotted) branch below collide in a SNIC bifurcation. Let x₀ = −1.6, the z-nullcline ( ) is at the Z-middle branch. The Z-upper (solid) and Z-lower (solid) branches correspond to DB and NS, respectively. The SLE attractor reduces then to a periodic transition between DB and NS. B, Deterministic time series of the Epileptor system ψ, subsystem 1 ψ₁, and subsystem 2 ψ₂. Parameters are: m = − 8, I_ext2 = 0, r  =  0.0005.

A, Top, Bifurcation diagram (z, x₁) of the Epileptor model. Parts I and II are zoomed on the bottom. The transitions between the Z-upper and Z-lower branches occur through a fold/circle bifurcation. B, Deterministic time series of the Epileptor system ψ, subsystem 1 ψ₁, and subsystem 2 ψ₂. Parameters are: m = −8, x₀ = −1.6, I_ext2 = 0.45, r  =  0.0005, I.C = [0 −5 2.7 0 0 0.01], and T_s = [0:0.001: 1600].

Stabilizing the equilibrium point of the nonoscillatory state and DB. The Epileptor model remains in the ictal nonoscillatory state after a fast discharges period. We plot a (z, x₁) bifurcation diagram for different values of m, x₀ and I_ext2. A–C, I_ext2 = 0.45, m  =  0 and x₀ = −0.9 ( A), I_ext2 = 0, m = −0.5, and x₀ = −0.8 ( B), and I_ext2 = 0, m = −1, and x₀ = −0.8 ( C). The equilibrium point C is a stable focus for A–C. The Epileptor stabilizes its equilibrium point C after transient seizure-like fast discharges, which disappear through a SNIC bifurcation ( A) and a through Hopf bifurcation ( B). The branches description for A–C is provided in Figures 5, 7, and 8, respectively. D, We plot a (z, x₁) bifurcation diagram with respect to z for m = −8 and I_ext2 = 0. The Z-upper (dashed), Z-middle (dash-dotted), and Z-lower (solid) branches consist of stable nodes, saddles, and stable nodes, respectively. Let x₀ = −0.6, the z-nullcline ( ) intersects the Z-upper branch at C, which is a stable node. The Epileptor model remains in DB after a transient NS. For all deterministic simulations: I.C = [−1 −5 4 0 0 0.01]. r  =  0.00095 and T_s = [0:0.001:2000] for A; r  =  0.002 and T_s = [0:0.001:2000] for B; r  =  0.001 and T_s = [0:0.001:2000] for C; and r  =  0.00005 and T_s = [0:0.001:3000] for D.

Stabilizing the equilibrium point of the NS. We plot a (z, x₁) bifurcation diagram with respect to z, as m and I_ext2 vary (x₀ = −2.5). A–F, m  =  0.5 and I_ext2 = 0.45 ( A), m  =  0 and I_ext2 = 0.45 ( B), m  =  0 and I_ext2 = 0 ( C), m = −0.5 and I_ext2 = 0 ( D), and m = −1 and I_ext2 = 0 ( E, F). Branches of the (z, x₁) curve in A are the same as in Figure 4. Branches of the (z, x₁) curve in B are the same as in Figure 5. Branches of the (z, x₁) curve in C are the same as in Figure 6. Branches of the (z, x₁) curve in D are the same as in Figure 7. Branches of the (z, x₁) curve in E and F are the same as in Figure 8. A–F, The z-nullcline ( ) intersects the Z-lower branch at C, which is a stable node. The Epileptor model remains in NS after a transient ictal state. For all (deterministic) simulations T_s = [0:0.001:4500]. I.C = [0.5 −5 2.9 0 0 0.01] and r  =  0.00053 for A; I.C = [0.5 −5 2.9 0 0 0.01] and r  =  0.00035 for B; I.C = [0 −5 2.8 0 0 0.01] and r  =  0.0007 for C; I.C = [0.5 −5 2.7 0 0 0.01] and r  =  0.00035 for D; I.C = [0.5 −5 2.7 0 0 0.01] and r  =  0.0005 for E; and I.C = [0.5 −5 2.45 0 0 0.01] and r  =  0.0004 for F.

A, Coexistence of SLEs and RSE. The simulations are performed without noise. SLEs and a stable LC coexist for m  =  0.5 and x₀ = −1.6 ( ). SLEs occur through a fold/homoclinic bifurcation. The arrows indicate the direction of trajectories. For easier visualization, we plot generalized coordinates (X, Y) corresponding to ( ) for seizures (top) and to ( ) for LC (bottom). LC is characteristic of RSE. B, C, Time series of SLEs ( B) and LC ( C). Parameter settings correspond to region VIII in Figure 31 and to region 13 in Figure 32. A, Top, I.C = [0 −5 3 0 0 0.01] and r  =  0.035. A, Bottom, I.C = [0 −5 −1 0 0 0.01] and r  =  0.035. The coexistence of LC and separatrix (S) can be found in area II [Fig. 12A].

A, Coexistence of SLEs and RSE. The simulations are performed without noise. SLEs and a stable LC coexist for m  =  0 and x₀ = −1.6 ( ). SLEs occur through a fold/homoclinic bifurcation. The arrows indicate the direction of trajectories. For easier visualization, we plot generalized coordinates (X, Y) corresponding to ( ) for seizures (top) and to ( ) for LC (bottom). LC is characteristic of RSE. B, C, Time series of SLEs ( B) and LC ( C). Parameter settings correspond to region X in Figure 34 and to region 15 in Figure 35. A, Top, I.C = [0 −5 3 0 0 0.01], T_s = [0 220], and r  =  0.01. A, Bottom, I.C = [4 −1 −0.5 0 0 0.01], T_s = [0 120], and r  =  0.01. The coexistence of LC and S can be found in area II (Fig. 12B).

A, Coexistence of SLEs and RSE. The simulations are performed without noise. SLEs and a stable LC coexist for m  =  0 and x₀ = −1.6 ( ). SLEs occur through a fold/circle bifurcation. The arrows indicate the direction of trajectories. For easier visualization, we plot generalized coordinates (X, Y) corresponding to ( ) for seizures (top) and to ( ) for LC (bottom). LC is characteristic of RSE. B, C, Time series of SLEs ( B) and LC ( C). Parameter settings correspond to region VII in Figure 31 and to region 12 in Figure 32. A, Top, I.C = [0 −5 3 0 0 0.01], T_s = [0 220], and r  =  0.01. A, Bottom, I.C = [4 −1 −0.5 0 0 0.01], T_s = [0 120], and r  =  0.01. The coexistence of LC and S can be found in area II (Fig. 12A).

A, Coexistence of SLEs and RSE. The simulations are performed without noise. SLEs and a stable LC coexist for m = −0.5 and x₀ = −1.6 ( ). SLEs occur through a fold/Hopf bifurcation. The arrows indicate the direction of trajectories. For easier visualization, we plot generalized coordinates (X, Y) corresponding to ( ) for seizures (top) and to ( ) for LC (bottom). LC is characteristic of RSE. B, C, Time series of SLEs ( B) and LC ( C). Parameter settings correspond to region IX in Figure 34 and to region 14 in Figure 35. A, Top, I.C = [0 −5 3 0 0 0.01], T_s = [0 220], and r  =  0.01. A, Bottom, I.C = [4 −1 −0.5 0 0 0.01], T_s = [0 120], and r  =  0.01. The coexistence of LC and S can be found in area II [Fig. 12B].

A, Coexistence of RSE and the periodic switch between nonoscillatory state and NS. The simulations are performed without noise. Here, the SLE attractor reduces to the periodic switch between nonoscillatory state and NS, which coexists with a stable LC for m = −1 and ( ). This periodic switch occurs through a fold/fold bifurcation. The arrows indicate the direction of trajectories. For easier visualization, we plot generalized coordinates (X, Y) corresponding to ( ) for the periodic switch between nonoscillatory state and NS (top) and to ( ) for LC (bottom). LC is characteristic of RSE. B, C, Time series of this periodic switch ( B) and LC ( C). Parameter settings correspond to region VIII in Figure 34 and to region 13 in Figure 35. A, Top, I.C = [0 −5 3 0 0 0.01], T_s = [0 220], and r  =  0.01. A, Bottom, I.C = [4 −1 0.5 0 0 0.01], T_s = [0 120], and r  =  0.01. The coexistence of LC and S can be found in area II (Fig. 12B).

A, Coexistence of SLC and RSE. The simulations are performed without noise. SLC and a stable LC coexist for m  =  1 and ( ). The equilibrium point is a saddle. The arrows indicate the direction of trajectories. For easier visualization, we plot generalized coordinates (X, Y) corresponding to ( ) for SLC (top) and to ( ) for LC (bottom). LC is characteristic of RSE. B, C, Time series of SLC ( B) and LC ( C). Parameter settings correspond to region II in Figure 34 and to region 17 in Figure 35. A, Top, I.C = [−1.15 −5 3.4 0 0 0.01], T_s = [0 500], and r  =  0.0035. A, Bottom, I.C = [10 −5 −1 0 0 0.01], T_s = [0 500], and r  =  0.0035.The coexistence of LC, S, and SLC can be found in area III [Fig. 12B].

A, Coexistence of SLC and RSE. The simulations are performed without noise. SLC and a stable LC coexist for m = −2 and ( ). The equilibrium point is an unstable focus. The arrows indicate the direction of trajectories. For easier visualization, we plot generalized coordinates (X, Y, Z) corresponding to ( ) for SLC (top) and to ( ) for LC (bottom). LC is characteristic of RSE. B, C, Time series of SLC ( B) and LC ( C). Parameter settings correspond to region II in Figure 34 and to region 2 in Figure 35. A, Top, I.C = [−1.15 −5 2.9 0 0 0.01], , and r  =  0.007. A, Bottom, I.C = [10 −5 −1 0 0 0.01], T_s = [0 500], and r  =  0.002. The coexistence of LC, S, and SLC can be found in area III [Fig. 12B].

A, Coexistence of RSE and a periodic switch between DB and NS. The simulations are performed without noise. Here, the SLE attractor reduces to the periodic switch between DB and NS, which coexists with a stable LC for m = −8 and x₀ = −1.4 ( ). Trajectory segments are numbered in A and B. DB corresponds to the segment 1, and the NS to the segment 3. The arrows indicate the direction of trajectories. For easier visualization, we plot generalized coordinates (X, Y) corresponding to ( ) for the periodic switch between DB and NS (top) and to ( ) for LC (bottom). LC is characteristic of RSE. B, C, Time series of the periodic switch between DB and NS ( B) and LC ( C). Parameter settings correspond to region VII in Figure 34 and to region 12 in Figure 35. A, Top, I.C = [0 −5 3 0 0 0.01], T_s = [0 450], and r  =  0.01. For easier visualization, we plot the trajectory over T_s = [96 244]. A, Bottom, I.C = [9 −5 −1 0 0 0.01], T_s = [0 200], and r  =  0.01. The coexistence of LC and S can be found in area II (Fig. 12B).

A, Coexistence of normal state and RSE. The simulations are performed without noise. The equilibrium point of NS and a stable LC coexist for m  =  0 and . Trajectory segments are numbered in A and B. The transient seizure-like fast discharges correspond to the segment 1, and the NS to the segment 4. The equilibrium point of NS exists. After the transient seizure-like fast discharges, the Epileptor remains in NS. The arrows indicate the direction of trajectories. For easier visualization, we plot generalized coordinates (X, Y) corresponding to ( ) for DB (top) and to ( ) for LC (bottom). LC is characteristic of RSE. B, C, Time series of NS ( B) and LC ( C). Parameter settings correspond to region X in Figure 31 and to region 18 in Figure 32. The coexistence of LC and S can be found in area II (Fig. 12A).

A, Coexistence of DB and RSE. The simulations are performed without noise. The equilibrium point of DB and a stable LC coexist for m = −8 and x₀ = −0.6. Trajectory segments are numbered in A and B. DB corresponds to the segment 4, and the NS to the segment 2. The equilibrium point of DB exists. The equilibrium point of NS does not. After a transient NS, the Epileptor stabilizes on DB. The arrows indicate the direction of trajectories. For easier visualization, we plot generalized coordinates (X, Y) corresponding to ( ) for DB (top) and to ( ) for LC (bottom). LC is characteristic of RSE. B, C, Time series of DB ( B) and LC ( C). Parameter settings correspond to region V in Figure 34 and to region 9 of 5 in Figure 35. A, Top, I.C = [−0.1 −6 3.8 0 0 0.01], T_s = [0 400], and r  =  0.01. A, Bottom, I.C = [9 −5 −1 0 0 0.01], T_s = [0 200], and r  =  0.009. The coexistence of LC and S can be found in area II (Fig. 12B).

A, Coexistence of nonoscillatory state and RSE. Here, transient seizure-like fast discharges disappears through a SNIC bifurcation, and then the nonoscillatory state occurs. The simulations are performed without noise. The equilibrium point of the nonoscillatory state and a stable LC coexist for m  =  0 and x₀ = −0.9 ( ). Trajectory segments are numbered in A and B. The transient seizure-like fast discharges correspond to the segment 2, the transient NS to the segment 1, and the nonoscillatory state to the segment 4. The equilibrium point of nonoscillatory state exists. After the transients NS and then seizure-like fast discharges, the Epileptor remains in the nonoscillatory state. The arrows indicate the direction of trajectories. For easier visualization, we plot generalized coordinates (X, Y) corresponding to ( ) for the nonoscillatory state (top) and to ( ) for LC (bottom). LC is characteristic of RSE. B, C, Time series of the nonoscillatory state ( B) and LC ( C). Parameter settings correspond to region V in Figure 31 and to region 6 in Figure 32. A, Top, I.C = [−1.5 −2.5 3.5 0 0 0.01], T_s = [0 500], and r  =  0.007. A, Bottom, I.C = [10 −5 −1 0 0 0.01], T_s = [0 500], and r  =  0.004. The coexistence of LC and S can be found in area II (Fig. 12A).

A, Coexistence of nonoscillatory state and RSE. Here a transient seizure-like fast discharges disappear through a Hopf bifurcation, and then the nonoscillatory state occurs. The simulations are performed without noise. The equilibrium point of the nonoscillatory state and a stable LC coexist for m = −0.5 and x₀ = −0.8 ( ). Trajectory segments are numbered in A and B. The transient seizure-like fast discharges correspond to the segment 2, the transient NS to the segment 1, and the nonoscillatory state to the segment 4. The equilibrium point of the nonoscillatory state exists. After the transients NS and then seizure-like fast discharges, the Epileptor remains in the nonoscillatory state. The arrows indicate the direction of trajectories. For easier visualization, we plot generalized coordinates (X, Y) corresponding to ( ) for the nonoscillatory state (top) and to ( ) for LC (bottom). LC is characteristic of RSE. B, C, Time series of the nonoscillatory state ( B) and LC ( C). Parameter settings correspond to region V in Figure 34 and to region 6 in Figure 32. A, Top, I.C = [−1 −5.5 3.8 0 0 0.01], T_s = [0 200], and r  =  0.02. A, Bottom, I.C = [10 −5 −1 0 0 0.01], T_s = [0 250], and r  =  0.004. The coexistence of LC and S can be found in area II (Fig. 12B).

A, Coexistence of nonoscillatory state and RSE. Here after a transient NS, the Epileptor enters into the nonoscillatory state. The simulations are performed without noise. The equilibrium point of the nonoscillatory state and a stable LC coexist for m = −1 and x₀ = −0.8 ( ). Trajectory segments are numbered in A and B. The transient NS is indicated by (1) and the nonoscillatory state (final state) by (3). The equilibrium point of nonoscillatory state exists, which is a stable focus. After a transient NS, the Epileptor spirals into the equilibrium point (stable focus) and remains in the nonoscillatory state. The arrows indicate the direction of trajectories. For easier visualization, we plot generalized coordinates (X, Y) corresponding to ( ) for the nonoscillatory state (top) and to ( ) for LC (bottom). LC is characteristic of RSE. B, C, Time series of the nonoscillatory state ( B) and LC ( C). Parameter settings correspond to region V in Figure 34 and to region 6 in Figure 32. A, Top, I.C = [−1 −5.5 3.5 0 0 0.01], T_s = [0 200], and r  =  0.01. A, Bottom, I.C = [10 −5 −1 0 0 0.01], T_s = [0 250], and r  =  0.004. The coexistence of LC and S can be found in area II (Fig. 12B).

Parameter space of the Epileptor model with respect to the parameters m and x₀ ( ). There are 12 regions separated by a SNPO bifurcation (bold line). LC exists above and it does not exist below. LC exists in area I. Area II shows bistability of LC and SLC. In area III, only SLC exists. Only a nonoscillatory state with damped oscillation exists in area IV, and coexists with LC in area V. Only SLE exists in area VI, and coexists with LC in areas VII and VIII. SLE occurs through a fold/circle bifurcation in areas VI and VII, and through a fold/homoclinic bifurcation in area VIII. NS exists in area IX and coexists with LC in area X. In areas XI and XII, LC coexists with a chaotic state which is periodic in area XI, and it does not in area XII.

The Epileptor parameter space of equilibrium points and periodic orbits with respect to the parameters m and x₀ ( ). There are 18 areas separated by a SNPO bifurcation (bold line). LC exists above and it does not exist below. The z-nullcline intersects the (z, x₁) curve at different equilibrium points. The equilibrium point is an unstable focus for areas 1 through 3. The equilibrium points are a stable node, an unstable focus and a saddle in areas 4 and 5. The equilibrium points are a stable focus, an unstable focus, and a saddle in area 6. The equilibrium points are one saddle and two unstable foci in area 7. The equilibrium points are a saddle, an unstable focus, and an unstable node in area 8. The equilibrium points are one unstable focus and two saddles in area 9. The equilibrium points are three saddles in area 10. The equilibrium point is a saddle for areas 11 through 16. The equilibrium point is a stable node in areas 17 and 18. In areas 1, 7, 8, 9, 10, and 14, only LC exists. Area 2 presents bistability of LC and SLC. In area 3, only SLC exists. In area 4, only a nonoscillatory state exists. In areas 5 and 6, LC and a nonoscillatory state coexist. Only SLE exists in area 11, and coexists with LC in areas 12 and 13. SLE occurs through a fold/circle bifurcation in areas 11 and 12, and through a fold/homoclinic bifurcation in area 13. NS exists in area 17 and coexists with LC in area 18. In areas 15 and 16, LC coexists with a chaotic state which is periodic in area 15, and it does not in area 16.

Interspike intervals ( ). A, B, m  =  1 ( A) and m  =  1.5 ( B).

Parameter space of the Epileptor model with respect to the parameters m and x₀ ( ). There are 13 regions separated by a SNPO bifurcation (bold line). LC exists above, and it does not exist below. LC exists in area I. Area II presents a bistability of LC and SLC. Only SLC exists in area III. DB exists for areas IV, V, VII, and VIII. In area IV, only DB exists. In area V, DB and LC coexist. In area VII, only a periodic switch between DB and NS exists. In area VIII, a periodic switch between DB and NS coexists with LC. Increasing m, DB locks into nonoscillatory state with damped oscillation coexisting with LC in area VI. A periodic switch between a nonoscillatory state and a NS coexists with LC in area IX. SLE coexists with LC in areas X and XI. SLE occurs through a fold/Hopf bifurcation in area X and through a fold/homoclinic bifurcation in area XI. NS exists in area XII and coexists with LC in area XIII.

The Epileptor parameter space of equilibrium points and periodic orbits with respect to the parameters m and x₀ ( ). There are 20 regions separated by a SNPO bifurcation (bold line). LC exists above and it does not exist below. The z-nullcline intersects the (z, x₁) curve at different equilibrium points. The equilibrium point is an unstable focus for areas 1 through 3. The equilibrium points are a stable node, an unstable focus, and a saddle in areas 4 and 5. The equilibrium points are a stable focus, an unstable focus, and a saddle in area 6. The equilibrium points are one saddle and two unstable foci in area 7. The equilibrium point is a stable node in areas 8, 9, 11, 19, and 20. The equilibrium point is a stable focus in area 10. The equilibrium point is a saddle for areas 12 through 18. In areas 1, 7, and 17, only LC exists. Areas 2 and 18 present bistability of LC and SLC. In area 3, only SLC exists. In areas 4 and 8, only DB exists. In areas 5 and 9, LC and DB coexist. In areas 6, 10, and 11, LC and nonoscillatory state coexist. Only a periodic switch between DB and NS exists in area 12, and coexists with LC in area 13. A periodic switch between a nonoscillatory state and NS coexists with LC in area 14. LC and SLE coexist in areas 15 and 16. SLE occurs through a fold/Hopf bifurcation in area 15 and through a fold/homoclinic bifurcation in area 16. NS exists in area 19 and coexists with LC in area 20.

A, Deterministic time series of the Epileptor model ψ varying m and x₀. a, a₁ , m = −12 and x₀ = −1.6. I_ext2 = 0.45 ( a) and I_ext2 = 0 ( a₁ ). The transitions between ictal and normal states occur through a fold/circle bifurcation ( a), and they reduce to transitions between DB and NS ( a₁ ). b, b₁ , m  =  0 and x₀ = −1.6. I_ext2 = 0.45 ( b) and I_ext2 = 0 ( b₁ ). The transitions between ictal and normal states occur through a fold/circle bifurcation ( b), and they reduce to transitions between nonoscillatory state and NS ( b₁ ). c, c₁ , x₀ = −1.6 and I_ext2 = 0. m = −1 ( c) and m = −0.5 ( c₁ ). The transitions between nonoscillatory state and NS occur through a fold/fold bifurcation ( c) and a fold/Hopf bifurcation ( c₁ ). r  =  0.0009 for ( a, a₁ ), r  =  0.001 for ( b, b₁ ), and r  =  0.0009 for ( c, c₁ ). B, Deterministic time series of the Epileptor model ψ, which show the Epileptor remaining in a nonoscillatory state, with m = −2, x₀ = −1 and ( a) in NS, with m = −2, x₀ = −2.5 ( b). r  =  0.001 for a and b.

A, B, Deterministic time series of the Epileptor model ψ, subsystem 1 ψ₁ and subsystem 2 ψ₂ with a chaotic spiking for m  =  1, x₀ = −1.6 ( A), and a chaotic transition between ictal and normal states for m  =  1.5, x₀ = −1.9 ( B).

Finding min(X) + 4 (red solid) and max(X) + 4 (black dashed) with respect to m. X is a solution of Equation 41. Blue (dotted) curve corresponds to . The equilibrium points of the subsystem 1 ( ) exist according to Equation 42.

The (x₁, y₁) phase plane of the subsystem 1. Possible intersections of x₁- (black cubic curve and dashed straight line ) and y₁- (green parabola) nullclines depending on z and m. Trajectories are plotted without noise starting from different initial conditions (black dot). The arrows indicate the direction of trajectories. The equilibrium points where they exist are labeled by red squares for stable nodes, and black stars for saddles. A, z  =  3.1, three equilibrium points coexist: a stable node (bottom), a saddle (middle), and a stable focus for m  =  0. The stable focus becomes unstable when m  =  1 and m  =  1.5, surrounded by a stable limit cycle. The limit cycle radius increases as m is increased. B, z  =  0, one equilibrium point exists which is an unstable focus surrounded by a stable limit cycle for m  =  0, and a stable focus for m = −10. C, m  =  1.5, one equilibrium point exists for z  =  2.7, and three equilibrium points exist for z  =  4: a stable node, a saddle, and an unstable focus. The stable limit cycle does not surround the unstable focus, and then is broken through a homoclinic bifurcation.

The (z, x₁) subsystem 1 bifurcation diagram with respect to z when m  =  0. On the left, the plot shows a Z-shaped curve . Z-lower (solid) and Z-middle (dash-dotted) branches consist of stable nodes and saddles, respectively. Z-upper (solid) sub-branch consists of stable foci and Z-upper (dashed) sub-branch consists of unstable foci. The two sub-branches are separated by a Hopf bifurcation H. The Z-lower and Z-middle branches collide as z decreases in a saddle-node bifurcation SN₁. The Z-upper and Z-middle branches collide as z increases in a saddle-node bifurcation SN₂. The Z-shaped curve comprises two branches : one (dashed) consists of unstable nodes and another (dash-dotted) consists of saddles. Decreasing z, they collide in a saddle-node bifurcation, SN.

A, B, The ( ) subsystem 1 bifurcation diagram with respect to z ( ), when m  =  0 ( A) and m  =  2 ( B). Z-lower (solid) and Z-middle (dash-dotted) branches consist of stable nodes and saddles, respectively. Z-upper branch consists of unstable foci ( B), and it is divided into two sub-branches separated by a Hopf bifurcation, H ( A): one (solid) consists of stable foci and another (dashed) consists of unstable foci. The Z-lower and Z-middle branches collide as z decreases in a saddle-node bifurcation SN₁. The Z-upper (solid for A and dashed for B) and Z-middle branches collide as z increases in a saddle-node bifurcation SN₂. A stable limit cycle ends at a Hopf bifurcation ( A) and at a homoclinic bifurcation ( B). The curves and correspond to the maximum, minimum, and averaged values along periodic orbits.

Finding bifurcation points with respect to m. SN₁ (blue dashed) and SN₂ (red dash-dotted) correspond to saddle-node bifurcations, and H (black solid) to a Hopf bifurcation. .

Finding the equilibrium points and periodic orbits of the fast-slow subsystem by using the (z, x₁) bifurcation diagram of the subsystem 1 shown in Figure 41. The z-nullcline ( ) intersects the Z-shaped curve at equilibrium points, and the curve at periodic orbits. A, B, m  =  0 and x₀ = −1.6 ( A), m  =  2 and x₀ = −1.6 (top, purple; B) or x₀ = −1.9 (bottom, red; B). LC and S are stable and saddle periodic orbits, respectively.

A, B, Finding equilibrium points of the fast-slow subsystem with respect to x₀, for m  =  0 ( A) and m  =  2 ( B). Blue (bottom solid), black (plus sign markers), red (dashed), and green (top solid) branches consist of stable nodes, saddles, unstable foci, and stable foci, respectively. SN₁ and SN₂ correspond to saddle-node bifurcation points and H to a Hopf bifurcation point.

Time series of the fast-slow subsystem as m and x₀ vary. Initial conditions are [0 −5 3] for left column and [0 −5 1] for right column. r  =  0.002. A, A₁ , m  =  0, x₀ = −1.6, transitions between upper and lower states occur through a fold/fold bifurcation for A; LC exists for A₁ . B, B₁ , m  =  2, x₀ = −1.6, only LC exists. C, C₁ , , SLC₁ exists for C and LC for C₁ . D, D₁ , m  =  2, , transitions between upper and lower states occur through a fold/homoclinic bifurcation for D; LC exists for D₁ . E, E₁ , , only a resting state exists. F, F₁ , m  =  0.5, , transitions between upper and lower states occur through a fold/Hopf bifurcation for F; LC exists for F₁ .

The ( ) subsystem 1 bifurcation diagram with respect to z as m and x₀ vary. A, m  =  0.5 and x₀ = −1.6, transitions between upper and lower branches occur through a fold/Hopf bifurcation. B, m  =  2 and x₀ = −1.9, transitions between upper and lower branches occur through a fold/homoclinic bifurcation. C, m  =  2 and x₀ = −1.7, the fast-slow subsystem stabilizes before a homoclinic bifurcation occurs, giving rise to the stable limit cycle SLC₁. D, m  =  0 and x₀ = −2.6, the fast-slow subsystem stabilizes its stable node, which is the equilibrium point of the NS. The z-nullcline and curve do not intersect, hence NS exists and LC does not. r  =  0.0006, I.C = [−0.5 −5 2.831], and T_s = [0:0.01: 1120] for A; r  =  0.002, I.C = [0 −5 2], and T_s = [0:0.01: 1300] for B; r  =  0.003, I.C = [−1.5 −5 4], and T_s = [0:0.01: 600] for C; and r  =  0.002, I.C = [0 −5 2], and T_s = [0:0.01: 1000] for D.

Periodic orbits of subsystem 1. A, Finding periodic orbits with respect to z (constant). We plot the graph of the Equation 65 on its right-hand side for different values of x₀ (from bottom to top; x₀ = 1, −0.25, −1.6, −2.4, −2.8) for m  =  0 ( a) and (x₀ = −1.6, −1.7, −1.9, −2.6, −2.8) for m  =  2 ( b). Stable and saddle periodic orbits are labeled as black circles and squares, respectively, which disappear as x₀ decreases through a SNPO bifurcation. B, The fast-slow subsystem parameter space of periodic orbits and equilibrium points with respect to x₀. a, m  =  0. b, m  =  2. Saddle periodic orbits, S, are labeled as black squares (with dot). Stable periodic orbits LC (bottom, red) and SLC₁ (top, blue) are labeled as (+) markers. Equilibrium points are labeled as dots. Blue (segment 1), black (segment 2), green (segment 3), and purple (segment 4) dots correspond to stable nodes, saddles, stable foci and unstable foci, respectively.

Bifurcation diagram of fast-slow subsystem with respect to z, as m and x₀ vary. A, m = –8, x₀ = −1.4, the fast-slow subsystem exhibits a periodic switch between DB and NS. B, , the fast-slow subsystem stabilizes its stable node which is the equilibrium point of NS. C, , the fast-slow subsystem stabilizes its stable focus, which is the equilibrium point of nonoscillatory state. D, , the fast-slow subsystem stabilizes its stable node, which is the equilibrium point of DB. r  =  0.0005, I.C = [0 −5 2.817], and T_s = [0:0.01: 3000] for A; r  =  0.001, I.C = [0 −5 2], and T_s = [0:0.01: 2000] for B; r  =  0.001, I.C = [−1.5 −5 4], and T_s = [0:0.01: 3500] for C; and r  =  0.0005, I.C = [−1.5 −5 4], and T_s = [0:0.01: 5000] for D.

Time series of the fast-slow subsystem as m and x₀ vary. Initial conditions are [0 −5 3] for left column and [0 −5 −1] for right column. A, A₁ , m  =  0, x₀ = −0.5, the fast-slow subsystem remains in the nonoscillatory state ( A), which coexists with LC ( A₁ ). B, B₁ , m  =  0, x₀ = −2.4, the fast-slow subsystem remains in the NS ( B), which coexists with LC ( B₁ ). C, C₁ , m = −8, x₀ = −1.4, the fast-slow subsystem switches between DB and NS ( C), coexisting with LC ( C₁ ). D, D₁ , m = −8, x₀ = −0.6, the fast-slow subsystem remains in DB ( D), which coexists with LC ( D₁ ). r  =  0.001 ( A, A₁ ), r  =  0.005 ( B), r  =  0.001 ( B₁ ), r  =  0.005 ( C), r  =  0.001 ( C₁ ), r  =  0.001 for ( D), and r  =  0.01 ( D₁ ).

Parameter space of the fast-slow subsystem with respect to the parameters m and x₀. There are 10 areas separated by a boundary (bold line), above which LC exists, and below it does not. For large values of m and x₀ (area I) only LC exists. The adjacent area II shows the bistability of LC and a stable focus. Both attractors are separated by a saddle periodic orbit. In the middle, LC coexists with the SLE attractor separated by a saddle periodic orbit. SLE occurs via a saddle-node/saddle-node bifurcation (area VI) and a saddle-node/homoclinic bifurcation (area VII). In area VIII, the HB is not completed and gives rise to another (coexisting) stable periodic orbit with a small amplitude. In area V, DB coexists with LC. In area IX, normal brain activity and LC coexist. In area X, only normal activity exists.

Special cases. A, B, Coexistence of LC and a stable focus even if a saddle periodic orbit does not exist. A, m  =  0, x₀ = −0.1. Parameters are as follows: r  =  0.002, I.C = [0 −5 2.83] on the right; I.C = [0 −5 1.5] on the left, and T_s = [0:0.01:600]. B, m  =  0, x₀ = −0.05. Parameters are as follows: r  =  0.002, black trajectory: I.C = [0 −5 2.83] and T_s = [0:0.01:600]; Blue trajectory: I.C = [0 −5 2.84] and T_s = [0:0.01:900].

Subsystem 2 dynamics. A, The (x₂, y₂) phase plane of the subsystem 2. Possible intersections of x₂- (cubic curve) and y₂-nullclines depending on . a, , one equilibrium point (stable node) exists, to which trajectory converges. b, , three equilibrium points coexist: a stable node, a saddle, and an unstable focus. Trajectory converges to the stable node. c, , one equilibrium point exists that is an unstable focus. Trajectory converges to a stable limit cycle. B, Bifurcation diagram of subsystem 2 with respect to . The lower (solid), middle (dash-dotted), and upper (dashed) branches of the S-shaped curve consist of stable nodes, saddles, and unstable foci, respectively. A branch of limit cycles (red) originates at a SNIC bifurcation. The curves above and below correspond to the maximum and minimum values along periodic orbits, respectively.