Double inverse stochastic resonance with dynamic synapses

We investigate the behavior of a model neuron that receives a biophysically realistic noisy postsynaptic current based on uncorrelated spiking activity from a large number of afferents. We show that, with static synapses, such noise can give rise to inverse stochastic resonance (ISR) as a function of the presynaptic firing rate. We compare this to the case with dynamic synapses that feature short-term synaptic plasticity and show that the interval of presynaptic firing rate over which ISR exists can be extended or diminished. We consider both short-term depression and facilitation. Interestingly, we find that a double inverse stochastic resonance (DISR), with two distinct wells centered at different presynaptic firing rates, can appear.

Figure 1

(a) ISR in the response of the postsynaptic neuron with static synapses. The mean firing rate $\nu$ is plotted vs the presynaptic firing rate $f$. Vertical dashed lines correspond to presynaptic firing frequencies where the firing behavior of the postsynaptic neuron exhibits transitions as follows: For $f<f_{IC}$, only the initial condition effect exists (region b). For $f_{IC}<f<f_T$, the trapping effect begins to appear (region c). For $f_T<f<f_K$, trapping occurs immediately and in all trials (region d). For $f>f_K$, the kickout effect increasingly dominates (regions e and f). (b)–(f) Sample raster plots of postsynaptic neuron spiking activity across 50 trials for different regions (marked with the corresponding letters and colors) of the ISR curve in panel (a). The panels are constructed by plotting a dot to indicate the occurrence of a postsynaptic spike at a given time in a given trial. The mean frequencies of the Poissonian presynaptic spike trains that generate the background activity are (b) $f=0.1\mathrm{Hz}$, (c) $f=1\mathrm{Hz}$, (d) $f=10\mathrm{Hz}$, (e) $f=50\mathrm{Hz}$, and (f) $f=200\mathrm{Hz}$.

Figure 2

Changes in the ISR behavior of a postsynaptic neuron that receives background activity through depressing synapses. The level of short-term depression at the synapses is controlled by the recovery time constant ${\tau }_{\mathrm{rec}}$. Here, ${\tau }_{\mathrm{fac}}=0\mathrm{ms}$, indicating the absence of synaptic facilitation. The figures show the average firing rate $\nu$ vs the presynaptic firing rate $f$ for relatively low (a) and high (b) levels of depression. Other synapse model parameters are as in Fig. 1.

Figure 3

The influence of both short-term depression and facilitation on ISR. (a) $\nu$ vs $f$ for various values of ${\tau }_{\mathrm{fac}}$ for a fixed level of STD at synapses defined by ${\tau }_{\mathrm{rec}}=100\mathrm{ms}$. (b) $\nu$ vs $f$ for various values of ${\tau }_{\mathrm{rec}}$ for a fixed level of STF at synapses defined by ${\tau }_{\mathrm{fac}}=1000\mathrm{ms}$. Other synapse model parameters are as in Fig. 1.

Figure 4

A qualitative depiction of the standard deviation ${\sigma }_I$ of the synaptic current versus the presynaptic firing rate $f$ for static (curve a), competing (curve b), and depressing (curve c) synapses. The vertical axis is schematic, and the three horizontal lines marked ${\sigma }_{IC}$, ${\sigma }_T$, and ${\sigma }_K$ separate regions [colored as in Fig. 1] that correspond to the different dynamic mechanisms described in the main text. The vertical dashed lines are the same as those in Fig. 1. Synaptic parameter sets for curve (a) ${\tau }_{\mathrm{rec}}=0\mathrm{ms}$, ${\tau }_{\mathrm{fac}}=0\mathrm{ms}$, curve (b) ${\tau }_{\mathrm{rec}}=100\mathrm{ms}$, ${\tau }_{\mathrm{fac}}=1000\mathrm{ms}$, and curve (c) ${\tau }_{\mathrm{rec}}=1000\mathrm{ms}$, ${\tau }_{\mathrm{fac}}=0\mathrm{ms}$. Other synapse model parameters are as in Fig. 1.

Figure 5

Effects of the baseline fraction of released neurotransmitter $\mathcal{U}$ on synaptic current fluctuations and the postsynaptic mean firing rate. (a) Variation of ${\sigma }_I$ as a function of $f$ for different values of $\mathcal{U}$ when only STD is present, with ${\tau }_{\mathrm{rec}}=1000\mathrm{ms}$ and ${\tau }_{\mathrm{fac}}=0\mathrm{ms}$. (b) Similar analysis as in panel (a) for the case of competing STD and STF, with ${\tau }_{\mathrm{rec}}=100\mathrm{ms}$ and ${\tau }_{\mathrm{fac}}=1000\mathrm{ms}$. Note that the three schematic horizontal lines marked ${\sigma }_{IC}$, ${\sigma }_T$, and ${\sigma }_K$ separate regions [as in Fig. 1] that correspond to the different dynamic mechanisms described in the main text. (c) Variation of $\nu$ as a function of $f$ for different values of $\mathcal{U}$ in the case of depressing synapses. This shows the transition from no ISR to a single-well ISR and then to a weak DISR as $\mathcal{U}$ increases. (d) Variation of $\nu$ as a function of $f$ for different values of $\mathcal{U}$. In this case, the plots show the transition from DISR to a single-well ISR when STD and STF are both present at synapses.

Figure 6

Panel (a) shows the bifurcation diagram of the Hodgkin-Huxley neuron as a function of the bias current $I_0$. Thick (thin) solid lines represent stable (unstable) fixed points, marked SFP and UFP. Solid (open) circles represent the minimum and maximum values of the voltage during stable (unstable) spiking behavior. These are marked SLC and ULC. Panel (b) shows a magnification of panel (a), revealing the multistable region.

Figure 7

The effect of the excitability control parameter $I_0$ on the behavior of ISR and DISR. (a) $\nu$ vs $f$ for different values of $I_0$ in the case of purely depressing synapses (${\tau }_{\mathrm{rec}}=1000\mathrm{ms}$ and ${\tau }_{\mathrm{fac}}=0\mathrm{ms}$) that induce ISR at $I_0\sim 6.8\mu \mathrm{A}/{\mathrm{cm}}^2$. (b) Similar analysis as in panel (a) for the case of competing STD and STF mechanisms (${\tau }_{\mathrm{rec}}=100\mathrm{ms}$ and ${\tau }_{\mathrm{fac}}=400\mathrm{ms}$) that result in DISR for $I_0\sim 6.8\mu \mathrm{A}/{\mathrm{cm}}^2$. Other synapse parameters are as in Fig. 1.