Diffusion dynamics of a conductance-based neuronal population

We study the spatiotemporal dynamics of a conductance-based neuronal cable. The processes of one-dimensional (1D) and 2D diffusion are considered for a single variable, which is the membrane voltage. A 2D Morris-Lecar (ML) model is introduced to investigate the nonlinear responses of an excitable conductance-based neuronal cable. We explore the parameter space of the uncoupled ML model and, based on the bifurcation diagram (as a function of stimulus current), we analyze the 1D diffusion dynamics in three regimes: phasic spiking, coexistence states (tonic spiking and phasic spiking exist together), and a quiescent state. We show (depending on parameters) that the diffusive system may generate regular and irregular bursting or spiking behavior. Further, we explore a 2D diffusion acting on the membrane voltage, where striped and hexagonlike patterns can be observed. To validate our numerical results and check the stability of the existing patterns generated by 2D diffusion, we use amplitude equations based on multiple-scale analysis. We incorporate 1D diffusion in an extended 3D version of the ML model, in which irregular bursting emerges for a certain diffusion strength. The generated patterns may have potential applications in nonlinear neuronal responses and signal transmission.

Figure 1

(a) Bifurcation diagram of the 2D ML oscillator with respect to the stimulus current $I$. The thick solid blue line indicates the stable equilibrium branch whereas the dotted blue line indicates the unstable equilibrium branch of the system. The stable and unstable limit cycles are denoted by solid cyan and dotted red lines, respectively. Points SH and SN represent the subcritical Hopf bifurcation and saddle-node bifurcation, respectively. (b) Nullclines are plotted for the deterministic uncoupled 2D ML model. The intersections of the $u$ nullcline and $v$ nullcline are the fixed points: the stable steady states (SS1 and SS2), unstable steady state (US), and saddle-node bifurcation (SN). The time series of the deterministic uncoupled 2D ML model for different regimes [marked by the vertical lines in (a)] of $I$ are (c) $I=0.052$ [vertical green line in the inset of (a)], (d) $I=0.054$ [vertical magenta line in the inset of (a)], and (e) $I=0.2$ (vertical black line), respectively.

Figure 2

Spatiotemporal plots of the 2D ML cable with 1D diffusion for (a)–(d) $I=0.052$, (e)–(h) $I=0.054$, and (i)–(l) $I=0.2$ and diffusion coefficients (a), (e), and (i) $D=0.0001$; (b), (f), and (j) $D=0.0005$; (c), (g), and (k) $D=0.0037$; and (d), (h), and (l) $D=0.5$. The color bar of all these spatiotemporal plots indicates the value of the membrane voltage $u$. Transient parts are also shown to understand the patterns clearly.

Figure 3

Time series of the end oscillator of the 2D ML cable with 1D diffusion. The external current stimulus is (a)–(d) $I=0.052$, (e)–(h) $I=0.054$, and (i)–(l) $I=0.2$. The values of the diffusion coefficients for all the panels are the same as in Fig. 2. We choose the end oscillator as a random node to show the temporal evaluation for each panel.

Figure 4

Characterization of patterns of the diffusively coupled 2D ML model: boundaries of the emergence of various structures (hexagons and stripes) for (a) $I=0.052$, (b) $I=0.054$, and (c) $I=0.2$. The dashed magenta line indicates the values of $\mu$ at (a) $I=0.052$ (${\mu }^{*}=27.4153$), (b) $I=0.054$ (${\mu }^{*}=28.508$), and (c) $I=0.2$ (${\mu }^{*}=108.2896$). The thick black line indicates the condition for the existence of hexagons whereas the thick blue and green lines indicate the boundary of the stability of stripes and $H_{\pi }$ hexagons, respectively. (d) Time integration of Eq. (24) for $I=0.2$.

Figure 5

Pattern formation in the 2D ML cable with 2D diffusion for (a)–(c) $I=0.052$, (d)–(f) $I=0.054$, and (g)–(i) $I=0.2$ and diffusion coefficients (a), (d), and (g) $D=0.001$; (b), (e), and (h) $D=0.05$; (c) and (f) $D=0.09$; and (i) $D=5$.

Figure 6

Time series of the end oscillator and spatial plot of the improved 3D ML cable with 1D diffusion. The diffusion coefficients $D$ are (a) and (d) $D=0.026$, (b) and (e) $D=0.4$, and (c) and (f) $D=0.7$.