Noise-enhanced coupling between two oscillators with long-term plasticity

Spike timing-dependent plasticity is a fundamental adaptation mechanism of the nervous system. It induces structural changes of synaptic connectivity by regulation of coupling strengths between individual cells depending on their spiking behavior. As a biophysical process its functioning is constantly subjected to natural fluctuations. We study theoretically the influence of noise on a microscopic level by considering only two coupled neurons. Adopting a phase description for the neurons we derive a two-dimensional system which describes the averaged dynamics of the coupling strengths. We show that a multistability of several coupling configurations is possible, where some configurations are not found in systems without noise. Intriguingly, it is possible that a strong bidirectional coupling, which is not present in the noise-free situation, can be stabilized by the noise. This means that increased noise, which is normally expected to desynchronize the neurons, can be the reason for an antagonistic response of the system, which organizes itself into a state of stronger coupling and counteracts the impact of noise. This mechanism, as well as a high potential for multistability, is also demonstrated numerically for a coupled pair of Hodgkin-Huxley neurons.

Figure 1

Schematic representation of the model; see Eq. (1).

Figure 2

(a) Plasticity function (5) with $A_1=1,A_2=0.5,{\tau }_1=0.5$, and ${\tau }_2=1.4$; (b) the corresponding PDDP dynamics for $\mathit{w}\left(t\right)\in (0,{\mathit{w}}_{max})$; see (8).

Figure 3

Time evolution of the coupling weights and phase differences of two oscillators (2) with the coupling function $g\left(\phi \right)=sin\left(\phi \right)$ and the PDDP rule (7); see Fig. 2. Plot (a) shows a unidirectionally coupled state with ${\mathit{w}}_1\approx {\mathit{w}}_{max}=1$ and ${\mathit{w}}_2\approx 0$, and (b) shows the coexisting desynchronized (and uncoupled) state with ${\mathit{w}}_{1,2}\approx 0$. For (a) and (b) the same parameters were used: $\mathrm{\Delta }\omega =0.1,\mu =0.01,\delta =0.001$, and plasticity parameters as given in Fig. 2.

Figure 4

Noise-induced switching between the desynchronized (uncoupled) regime and the unidirectionally coupled phase-locked regime. Plot (a) shows a single event for $\delta =0.005$ at time $t_0\approx 900$. Switch events were detected as the first time $t_0$ for which the system reaches a state with ${\mathit{w}}_1\left(t_0\right)>0.5$. In (b) the mean switching time from the uncoupled state to the unidirectionally coupled state is shown in dependence of the plasticity rate $\delta$ for $\mu =0.01,0.1,0.2,$ and $0.3$. Other parameters as in Fig. 3.

Figure 5

Panels (a) and (b): Lines show stationary distributions ${\rho }_s\left(\phi \right)$ given by Eq. (21) in the case of fixed unidirectional coupling ${\mathit{w}}_1=1,{\mathit{w}}_2=0$. Bars show numerically obtained histograms from the computed phase dynamics (2). The histograms consist of 100 bins, and are based upon data from a simulation over $10000$ units of time. Panels (c) and (d): Stationary distributions ${\rho }_s$ for different noise intensities $\mu$ (black lines) and fixed unidirectional coupling. The red line indicates the value of the advection part $v\left(\phi \right)$ of the phase dynamics; see (17). Panels (a) and (c): results for the coupling function $g\left(\phi \right)=sin\left(\phi \right)$ and parameters $\mathrm{\Delta }\omega =0.1,\mu =0.2$. Panels (b) and (d): $g\left(\phi \right)=0.2sin\left(\phi \right)+cos\left(2\phi \right)$ and $\mathrm{\Delta }\omega =0.2,\mu =0.2$.

Figure 6

Phase portraits and bifurcation diagram for system (18) with $g\left(\phi \right)=sin\phi$ and $h\left(\phi \right)$ as in Fig. 2. Panels (a), (c): uncoupled and bidirectionally coupled states are stable; (b) uncoupled, unidirectionally and bidirectionally coupled states are stable; (d) uncoupled and unidirectionally coupled states are stable; (e), (f) only the uncoupled state is stable. A bold arrow between two phase portraits indicates that these regimes are connected by a bifurcation [solid lines in (g)]. Panel (g) shows the corresponding bifurcation diagram in the $(\mathrm{\Delta }\omega ,\mu )$ plane. Blue-shaded region: parameters, for which the bidirectional coupling $\left({\mathit{w}}_1={\mathit{w}}_2={\mathit{w}}_{max}=1\right)$ is stable [cf. (a)–(c)]. Red-shaded region: parameter for which the unidirectional coupling $\left({\mathit{w}}_1={\mathit{w}}_{max}=1,{\mathit{w}}_2={\mathit{w}}_{min}=0\right)$ is stable [cf. (b) and (d)]. For all parameters $\mathrm{\Delta }\omega \ne 0$ and $\mu$, the uncoupled state ${\mathit{w}}_1={\mathit{w}}_2=0$ is stable.

Figure 7

Three realizations of the stochastic system (1) with (8)–(10) starting from the same initial condition for the coupling weights $({\mathit{w}}_1\left(0\right),{\mathit{w}}_2\left(0\right))=(0.6,0.92)$ and converging either to a bidirectional coupling (trajectory 1) or to unidirectional coupling (trajectories 2 and 3). Parameters $\mathrm{\Delta }\omega =0.05$ and $\mu =1.5$, which correspond to the case (c) in Fig. 6.

Figure 8

Noise-induced coupling regimes of two interacting HH neurons (25) with random input vs parameters $\mathrm{\Delta }I$ and $\mu$. (a) The shaded domains correspond to parameter regions where different coupling configurations are stable. The stability region of the unidirectional coupling $({\mathit{w}}_1,{\mathit{w}}_2)\approx (0.5,0.0)$ is shaded in red, and the bidirectional coupling $({\mathit{w}}_1,{\mathit{w}}_2)\approx (0.5,0.5)$ in blue as in Fig. 6. The uncoupled state $({\mathit{w}}_1,{\mathit{w}}_2)\approx (0.0,0.0)$ is stable in the gray shaded region, and the stability region of the inverse unidirectional coupling $({\mathit{w}}_1,{\mathit{w}}_2)\approx (0.0,0.5)$ is indicated by green shading. The level curve $〈{\mathit{w}}_2〉=0.25$ with ${\mathit{w}}_1\approx 0.5$ is depicted by blue dashed curve. The vertical magenta dashed line indicates a cross section of the parameter plane used in Figs. 9 and 10. (b) The values of the time-averaged synaptic weight $〈{\mathit{w}}_2〉$ observed in a stable configurations with ${\mathit{w}}_1\approx 0.5$ are encoded in color.

Figure 9

Mean switching time of (a) uncoupled state, (b) inverse unidirectional coupling, and (c) bidirectional and unidirectional coupling vs input intensity $\mu$ for the fixed parameter of detuning $\mathrm{\Delta }I=0.05$ indicated by the magenta dashed line in Fig. 8. The vertical dashed lines in the plots indicate the border values of $\mu$ of stability domains [i.e., the intersection of the vertical magenta dashed line in Fig. 8 with the borders of the corresponding stability regions]. In plot (c) the value $\mu =0.049$ of the intersection of the vertical magenta dashed line $\mathrm{\Delta }I=0.05$ with the level curve $〈{\mathit{w}}_2〉=0.25$ in Fig. 8 is indicated by black dashed line instead. The insets show plots in linear-log scale. The mean switching time was obtained by averaging over 200 trajectories initiated at the corresponding coupling regime. The maximal simulation time was $300000\mathrm{s}$.

Figure 10

(a)–(d) Basins of attraction of coexisting coupling regimes of HH neurons (25) depicted by color in the plane of the initial synaptic weights $({\mathit{w}}_1\left(0\right),{\mathit{w}}_2\left(0\right))$ for (a) $\mu =0.1$, (b) $\mu =0.132$, (c) $\mu =0.137$, and (d) $\mu =0.15$ and fixed $\mathrm{\Delta }I=0.05$; see Fig. 8 (markers on the vertical magenta dashed line). The corresponding coupling regimes are indicated in the plots. Black dot and crosses in (c) indicate initial synaptic weights used for (e)–(h). Any fixed initial synaptic weight was assigned to the basin of attraction of that coupling regime to which the most of 31 different trajectories have been attracted after skipped transient. (e)–(h) Examples of the time courses of synaptic weights ${\mathit{w}}_1\left(t\right)$ (red curves) and ${\mathit{w}}_2\left(t\right)$ (blue curves) in system (25) for $\mu =0.137$ [as in (c)], fixed initial synaptic weights $({\mathit{w}}_1\left(0\right),{\mathit{w}}_2\left(0\right))=(0.17,0.4)$ [black dot in (c)], and random initial conditions for neurons' variables. The coupling regimes reached by the neurons after transient are indicated in the plots.

Figure 11

Time courses of the synaptic weights ${\mathit{w}}_1$ and ${\mathit{w}}_2$ in system (25) for (a) input-free case with $\mu =0$ and (b) neurons perturbed by an independent random synaptic input of intensity $\mu =0.03$ and different parameter mismatches $\mathrm{\Delta }I$ as indicated in the legends. As initial state, the uncoupled regime ${\mathit{w}}_1\left(0\right)={\mathit{w}}_2\left(0\right)=0.0$ was taken. Plots (c) and (d) show the time-averaged synaptic weight $\langle {\mathit{w}}_2\rangle$ in system (25) vs (c) parameter mismatch $\mathrm{\Delta }I$ for the input-free case $\mu =0$, (d) input intensity $\mu$ for different $\mathrm{\Delta }I$ as indicated in the legend. The other synaptic weight ${\mathit{w}}_1\approx {\mathit{w}}_{\max }=0.5$ [cf. (a) and (b)].

Figure 12

Neuronal dynamics in the input-free $\left(\mu =0\right)$ system (25) for (a) and (b) fixed synaptic weights (STDP is off) $\left({\mathit{w}}_1,{\mathit{w}}_2\right)=\left(0.5,0.5\right)$ and $\left({\mathit{w}}_1,{\mathit{w}}_2\right)=\left(0.5,0.0\right)$, respectively, and (c) adaptive synaptic weights (STDP is on) approaching the state $\left({\mathit{w}}_1,{\mathit{w}}_2\right)=({\mathit{w}}_{\max },w{}_2^{*})$ with ${\mathit{w}}_2^{*}\approx 0.21$ due to STDP; see Fig. 11 for $\mathrm{\Delta }I=0.02$. In plots (a) and (b) the time courses of the membrane potentials $V_j$ of the neurons are shown, and (c) depicts the distribution density of the spiking time difference $\mathrm{\Delta }t=t_{\mathrm{post}}-t_{\mathrm{pre}}$ for the postsynaptic neuron 2 contributing to the update of synaptic weight ${\mathit{w}}_2$. (d) Dynamics of the average update $〈\mathrm{\Delta }{\mathit{w}}_2〉$ (see text for definition) for the conditions of plots (a)–(c), as indicated in the legend. (e) Log-log plot of $〈\mathrm{\Delta }{\mathit{w}}_2〉$ for adaptive synaptic weights [blue circles in plot (d), corresponding to conditions of plot (c) and Fig. 11 for $\mathrm{\Delta }I=0.02]$. The black dashed line has a slope $-1$ and is given for comparison. Parameters $\mathrm{\Delta }I=0.02$ and $\mu =0$.

Figure 13

Impact of the random input on the neuronal dynamics of system (25). (a) Distribution densities of the spiking time difference $\mathrm{\Delta }t=t_{\mathrm{post}}-t_{\mathrm{pre}}$ for the postsynaptic neuron 2 contributing to the update of synaptic weight ${\mathit{w}}_2$ for the fixed coupling (STDP is off) $\left({\mathit{w}}_1,{\mathit{w}}_2\right)=(0.5,0)$ (blue empty histogram) and $\left({\mathit{w}}_1,{\mathit{w}}_2\right)=(0.5,0.5)$ (red hatched histogram) and input intensity $\mu =0.03$. The distribution of $\mathrm{\Delta }t$ for $\left({\mathit{w}}_1,{\mathit{w}}_2\right)=(0.5,0)$ and vanishing input $\left(\mu =0\right)$ is depicted by black filled histogram (scaling on the right vertical axis). (b) Dynamics of the average update $〈\mathrm{\Delta }{\mathit{w}}_2〉$ for the two conditions $\left({\mathit{w}}_1,{\mathit{w}}_2\right)=(0.5,0)$ and $\left({\mathit{w}}_1,{\mathit{w}}_2\right)=(0.5,0.5)$ of plot (a) with random input and fixed coupling (blue squares and red triangles) and adaptive synaptic weights (green circles) as indicated in the legend. (c) Log-log plot of $〈\mathrm{\Delta }{\mathit{w}}_2〉$ for the case when STDP is on [green circles in plot (b), corresponding to conditions of Fig. 11 for $\mathrm{\Delta }I=0.04]$. The black dashed line has a slope $-1$ and is given for comparison. (b) Distribution density of $\mathrm{\Delta }t$ for adaptive synaptic weights (STDP is on), where $\left({\mathit{w}}_1,{\mathit{w}}_2\right)$ fluctuates around $\left({\mathit{w}}_{\max },{\mathit{w}}_2^{*}\right)$ with ${\mathit{w}}_2^{*}\approx 0.2$ [Fig. 11 for $\mathrm{\Delta }I=0.04]$ corresponding to the case depicted by green circles in plots (b) and (c). Parameters $\mathrm{\Delta }I=0.04$ and $\mu =0.03$.

Figure 14

Time-averaged synaptic weights in system (25) for the multiplicative update rule with soft bounds vs (a) parameter mismatch $\mathrm{\Delta }I$ for the input-free case $\mu =0$ and (b) and (c) input intensity $\mu$ for different $\mathrm{\Delta }I$ as indicated in the plots. (d) Time courses of the membrane potentials of the HH neurons (25) for $\mathrm{\Delta }I=0.02$, noise intensity $\mu =0.2$, and coupling regime from plot (b). Parameter $V_r=0$ mV.