Transient chaos and associated system-intrinsic switching of spacetime patterns in two synaptically coupled layers of Morris-Lecar neurons

Spatiotemporal chaos collapses to either a rest state or a propagating pulse solution in a single layer of diffusively coupled, excitable Morris-Lecar neurons. Weak synaptic coupling of two such layers reveals system intrinsic switching of spatiotemporal activity patterns within and between the layers at irregular times. Within a layer, switching sequences include spatiotemporal chaos, erratic and regular pulse propagation, spontaneous network wide neuron activity, and rest state. A momentary substantial reduction in neuron activity in one layer can reinitiate transient spatiotemporal chaos in the other layer, which can induce a swap of spatiotemporal chaos with a pulse state between the layers. Presynaptic input maximizes the distance between propagating pulses, in contrast to pulse merging in the absence of synapses.

Figure 1

(a) A typical excitation cycle in phase space for a single ML neuron (thick dashed line) and a typical trajectory of a coupled neuron with chaotic dynamics (thin dashed line). The cubic $V$-nullcline and the $n$-nullcline (full lines) are shown, together with the three steady states, a stable node (diamond), a saddle point (circle), and an unstable focus (plus). Typical examples of spatiotemporal dynamics of the membrane potential ($V$) for a diffusively coupled ML network ($g=0, N=50$) when spatiotemporal chaos collapses to (b) the global rest state, and to (c) a traveling pulse solution. A membrane potential ($V$) close to the rest state (unstable focus) is represented in dark (lighter) color. The plotted time interval reflects the dynamics near the time of collapse. The simulations were started with a superthreshold perturbation $(V,n)=(-10\text{mV},0)$ of a single neuron, with all other neurons in the rest state. More details on generic initial conditions that reach the neighborhood of the chaotic saddle are presented in Keplinger et al. [17]. (d) Comparison of narrow (dashed line) and wide (solid line) pulse solution via their time evolution of the membrane potential $V$. The phase portraits of these two pulses are depicted in Fig. 8.

Figure 2

Typical system intrinsic switching sequences of spatiotemporal patterns in layers $\mathrm{I}$ and $\mathrm{II}$, initially in the STC state. The synaptic coupling is one-way, from layer $\mathrm{I}$ to $\mathrm{II}$, with standard coupling, $g=0.01 {\mathrm{mS}/\mathrm{cm}}^2$. Panels (a) and (b) differ by randomly chosen initial conditions. The vertical lines indicate a break in time. Times are given only to illustrate the time scale within a pattern. For other parameters, see Fig. 1.

Figure 3

Summary sketch for switching behavior in the case of one-way synaptic coupling from layer $\mathrm{I}$ to $\mathrm{II}$ and synapse strength, $g=0.01 {\mathrm{mS}/\mathrm{cm}}^2$. The notation [A, B] refers to state A in layer $\mathrm{I}$ and state B in layer $\mathrm{II}$ with following abbreviations C (STC), WP (wide pulse), NP (narrow pulse), EP (erratic pulse), S (stripe pattern), and R (rest state). Final states are asymptotic, and arrows mark state transitions. The collapse to multiple pulses is ignored.

Figure 4

Maximum membrane potential, $max\left(V\right)$, of a postsynaptic neuron (layer $\mathrm{II}$) versus synaptic coupling strength $g$, if the input layer of $N=50$ neurons exhibits spatiotemporal chaos (A), two traveling narrow pulses (B), a single traveling wide pulse (C), or a single traveling narrow pulse (D). The critical coupling strengths for the excitation of a postsynaptic neuron from the rest state are marked as vertical lines; they are clearly below $g=1.5 {\mathrm{mS}/\mathrm{cm}}^2$, for which a single active presynaptic neuron can excite a postsynaptic neuron in the rest state and thus trivially keep layer $\mathrm{II}$ active.

Figure 5

(a) Spatiotemporal chaos in layer $\mathrm{I}$, and (b) corresponding stripe locations in layer $\mathrm{II}$ as a function of synaptic coupling $g$. (c) The rate $R_s$ (number of stripes per 200 s) at which a chaotic input layer generates stripes in the output layer as a function of coupling strength $g$. Stripe patterns are absent for stronger synaptic coupling, since neurons in layer $\mathrm{II}$ are continuously excited.

Figure 6

Time series of presynaptic input ($I_p=g{\sum }_{j=1}^N{s}_{ji}^{\mathrm{I},\mathrm{II}}$, dashed line), synaptic current ($I_i^{\mathrm{syn},\mathrm{II}}$, full line), and membrane potential ($V_i^{\mathrm{II}}$, dotted line) of postsynaptic neuron ($i$) for the simulation in Fig. 5 with synapse strength $g=0.01 {\mathrm{mS}/\mathrm{cm}}^2$. The horizontal lines mark the zero line and the membrane potential in the rest state.

Figure 7

(a) Spatiotemporal chaos in layer $\mathrm{I}$ and (b) corresponding erratic pulse pattern in layer $\mathrm{II}$. (c) The stripe pattern (Sec. 3a) initiated from layer $\mathrm{I}$ is plotted for comparison. The synapse strength is $g=$ 0.01 ${\mathrm{mS}/\mathrm{cm}}^2$.

Figure 8

Phase space trajectories of a narrow pulse (full line), a (driven) narrow pulse (dashed line), a wide pulse (bold full line), a (driven) wide pulse (circles), and an erratic pulse (light colored full line). The narrow and wide pulse refer to pulse solutions in a diffusive layer without synaptic input [see also Fig. 1]. The (driven) narrow/wide pulse experiences synaptic input from layer $\mathrm{I}$ ($g=0.01{\mathrm{mS}/\mathrm{cm}}^2$), in this case from a narrow pulse in layer $\mathrm{I}$; the (driven) narrow/wide pulse becomes the narrow/wide pulse upon termination of the synaptic input. The erratic pulse develops from any pulse that experiences synaptic input from STC; the erratic pulse becomes a narrow pulse upon termination of the synaptic input.

Figure 9

Spatiotemporal chaos in (a) layer $\mathrm{I}$ and (b) layer $\mathrm{II}$. (c) An example for STC collapsing to four erratic pulses in layer $\mathrm{II}$. The synaptic strength is $g=$ 0.01 ${\mathrm{mS}/\mathrm{cm}}^2$.

Figure 10

(a) Largest Lyapunov exponent $\lambda$ [26, 46] and (b) average lifetime $\langle T\rangle$ of transient STC in layer $\mathrm{II}$ versus synaptic coupling strength $g$ when layer $\mathrm{I}$ is in the STC state. Error bars (one standard deviation) are plotted where scale allows; half standard deviation is marked for the last data point ($+$). The average is over 50 simulations with different initial conditions, a combination of 10 (5) randomly chosen STC initial conditions in layer $\mathrm{II}$ ($\mathrm{I}$). The distribution of lifetimes for each data point (not plotted) appears exponential, consistent with other studies [12, 16].

Figure 11

Spatiotemporal chaos in layer $\mathrm{I}$ (a) synaptically influences the double pulse pattern in layer $\mathrm{II}$ for a synapse strength of $g=$ 0.004 ${\mathrm{mS}/\mathrm{cm}}^2$ (b) and for the standard value $g=$ 0.01 ${\mathrm{mS}/\mathrm{cm}}^2$ (c). The two pulses in layer $\mathrm{II}$ result from the collapse of STC to a double pulse.

Figure 12

Time evolution of pulse distance $d_P$ (in layer $\mathrm{II}$) as a response to STC input from layer $\mathrm{I}$ for varying synapse strengths $g$ ($N=50$ neurons; Fig. 11). (a) The pulse distance for each coupling strength was calculated from interpolation of pulse peaks over a time interval of 5 ms, and then averaged for nine different STC realizations in layer $\mathrm{I}$. No change in pulse distance is observed for $g\in \{0.001,0.002,0.003,0.004\} {\mathrm{mS}/\mathrm{cm}}^2$; only the curve for $g=0.004{\mathrm{mS}/\mathrm{cm}}^2$ is included. (b) Long-term simulation over $100\mathrm{s}$ for a case of slow pulse distance maximization ($g=$ 0.006 ${\mathrm{mS}/\mathrm{cm}}^2$) and for the case of pulse merging in the absence of synaptic input ($g=0$).

Figure 13

Simultaneous change of dynamics in layers $\mathrm{I}$ and $\mathrm{II}$ ($g=$ 0.01 ${\mathrm{mS}/\mathrm{cm}}^2$): the collapse of spatiotemporal chaos to a wide pulse in layer $\mathrm{I}$ (a), and reinitiation of spatiotemporal chaos from the erratic pulse pattern in layer $\mathrm{II}$ (b).

Figure 14

Reinitiation of STC in layer $\mathrm{II}$ from an erratic double pulse (a) and an erratic quadruple pulse (b), analogous to Fig. 13. Spatiotemporal dynamics in layer $\mathrm{II}$ (c) and the corresponding symmetry parameter $P_s={\sum }_{i=1}^{25}\parallel V_i-V_{i+25}\parallel$ (d) after chaos reinitiation from a distance maximized double pulse in a layer of $N=50$ neurons and $g=$ 0.01 ${\mathrm{mS}/\mathrm{cm}}^2$.

Figure 15

A rare event of chaos reinitiation from an erratic pulse in layer $\mathrm{II}$ (b) through the synaptic STC input from layer $\mathrm{I}$ (a) at $g=0.01 {\mathrm{mS}/\mathrm{cm}}^2$, observed once in about 200 simulations.

Figure 16

Chaos reinitiation from a (driven) narrow pulse in layer $\mathrm{II}$ (right pattern) through the synaptic input of STC in layer $\mathrm{I}$ (left pattern) for tiny synaptic coupling strength: (a) $g=0.003 {\mathrm{mS}/\mathrm{cm}}^2$ (case A), (b) $g=0.0025 {\mathrm{mS}/\mathrm{cm}}^2$ (case B), and (c) $g=0.002 {\mathrm{mS}/\mathrm{cm}}^2$ (case C). The synaptic coupling is small enough that the pulse in layer $\mathrm{II}$ is not erratic, traveling with a rather constant speed. Only a fraction (2 s) of the simulation time near the time of chaos reinitiation is shown. (d) Presynaptic input ($I_p=g{\sum }_{j=1}^N{s}_{ji}^{\mathrm{I},\mathrm{II}}$) near the time of chaos reinitiation. The time interval represents about 1.5 s before and about 0.5 s after chaos reinitiation, slightly off centered within the box for clarity. The STC dynamics in layer $\mathrm{I}$ is identical for all three cases, and chaos reinitiation occurs after 33.7 s of simulation time for case A, after 7.3 s for case B, and after 3.6 s for case C. While the particular value of reinitiation time depends on the STC realization, increasing reinitiation times reflect that the required reduction in presynaptic input is more rare. The reduction of $I_p$ when STC collapses to a wide pulse in $\mathrm{I}$ (Sec. 3e, $g=0.01$ ${\mathrm{mS}/\mathrm{cm}}^2$) is shown for comparison (case D).

Figure 17

Summary sketch ($g$ = 0.01 ${\mathrm{mS}/\mathrm{cm}}^2$) for switching behavior in the case of two-way synaptic coupling between layers $\mathrm{I}$ and $\mathrm{II}$. Arrows mark transitions to states [A,B], and dashed arrows to symmetrically equivalent states [B,A]; e.g., [C,EP] is mapped onto [EP,C]. Further information in Fig. 3.

Figure 18

Spatiotemporal dynamics for two-way synaptic coupling ($g=$ 0.01 ${\mathrm{mS}/\mathrm{cm}}^2$) between layers $\mathrm{I}$ and $\mathrm{II}$. Example for (a) a swapping event, (b) a simultaneous chaos reinitiation, and (c) a fast switching sequence.