Figure 1

Comparison of the ${\text{IF}}_{\mathrm{g}}$ and ${\text{IF}}_{\mathrm{I}}$ models of conductance and current-based synaptic drive. Synaptic input is at a rate ${\mathcal{R}}_e=15.0$ and ${\mathcal{R}}_i=9.23\text{kHz}$. The ${\text{IF}}_{\mathrm{g}}$ has postsynaptic potentials (PSPs) with voltage-dependent amplitude, whereas those of the ${\text{IF}}_{\mathrm{I}}$ are fixed such that at the initial equilibrium voltage of $-65\text{mV}$ both models have PSPs with amplitudes of $0.13\text{mV}$ (${\tilde{a}}_e=0.002$,${\tilde{a}}_i=0.013$). In panels a and c the response to a step current of magnitude $I_{step}∕C=0.2\text{nA}∕\text{nF}$ with onset at $100\text{ms}$ is plotted: $(-)$ example time courses (with identical spike-train inputs) of the voltage $V$, $(-)$ mean voltage, $(\circ )$ simulations. (a) The mean voltage in the ${\text{IF}}_{\mathrm{g}}$ shifts by $1\text{mV}$, with a time constant of $\tau =5\text{ms}$. (c) The shift for the ${\text{IF}}_{\mathrm{I}}$ is $4\text{mV}$ with a time constant of ${\tau }_L=20\text{ms}$. Panels (b) and (d): the voltage distributions before (gray) and long after (black) the onset of the step current. The standard deviation of the ${\text{IF}}_{\mathrm{g}}$ model is $1.0\text{mV}$ whereas for the ${\text{IF}}_{\mathrm{I}}$ it is $2.0\text{mV}$.Reuse & Permissions

Figure 2

Response of the conductance-based synapses model ${\text{IF}}_{\mathrm{g}}$ to balanced changes [see Eq. (21)] in the presynaptic rates: a demonstration of the competing effects of synaptic fluctuations and synaptic conductance increase. (a) A balanced step change (at $100\text{ms}$) in the incoming rates (given in kHz the panel) with $E$ held constant at $-60\text{mV}$. At depolarized potentials an increased synaptic drive leads to an increase in the standard deviation from $1.35$ to $1.77\text{mV}$: the fluctuation increase dominates over the conductance increase. (b) The membrane voltage distributions before (gray) and long after (black) the change in synaptic drive. A weak skew in the distributions is visible. (c) A balanced step change in the incoming rates (given in the panel) with $E$ held constant at $-73\text{mV}$. At hyperpolarized potentials the increasing synaptic drive leads to a decrease in the standard deviation from $0.97$ to $0.60\text{mV}$: the conductance increase dominates over the increase in synaptic fluctuations. (d) The effect is seen in a sharpening of the distribution from before (gray) to after (black) the step change. In both panels (a) and (c) a number of example trajectories have been plotted in gray and the membrane time scales given before and after the step change. (e) The standard deviation as a function of increasing fluctuations (balanced synaptic drive parameterized by ${\mathcal{R}}_e$) for different values of fixed $E$ given in the legend. The initial point in each curve corresponds to ${\mathcal{R}}_i=0$ and the final to ${\mathcal{R}}_e=10\text{kHz}$. The time labels give the values of $\tau$ at these points (the inverse of which are related to the total conductance). The examples given in panels (a) and (c) can be identified on panel (e). The effect of increasing synaptic drive reverses at a mean membrane voltage of $E^{*}\simeq -68\text{mV}$ [see Eq. (24)]. In all cases ${\tilde{a}}_e=0.004$ and ${\tilde{a}}_i=0.026$ giving postsynaptic potentials of size $\simeq 0.26\text{mV}$ at $-65\text{mV}$. Simulational data are denoted by symbols.Reuse & Permissions

Figure 3

(a) Firing rate with increasing equilibrium potential $E$. The initial values of ${\mathcal{R}}_e=9.17$ and ${\mathcal{R}}_i=3.08\text{kHz}$ give $E=-60\text{mV}$ and a time constant of $6\text{ms}$. The equilibrium potential $E$ is then increased through three different mechanisms (see legend and text). (b) Firing-rate curves for balanced synaptic input (at fixed $E$ given in the legend) as a function of increasing presynaptic noise parametrized by ${\mathcal{R}}_e$. (c) Firing rate curves at constant conductance (both $E$ and $\tau =5\text{ms}$ fixed) as a function of increasing synaptic noise parameterized by the postsynaptic potential (PSP) size. This scenario corresponds to an increase in presynaptic synchrony and is achieved by keeping the products ${\mathcal{R}}_e{\tilde{a}}_e$ and ${\mathcal{R}}_i{\tilde{a}}_i$ fixed while ${\tilde{a}}_e$ and ${\tilde{a}}_i$ are varied. Here the EPSPs and IPSPs amplitudes are identical at $-65\text{mV}$ and provide the abscissa of the plot. The range of PSPs for which the diffusion approximation is valid is seen in the fit with simulations. In all figures the bold lines are the firing rate Eq. (19) and the broken lines a current-based approximation accounting for the tonic conductance increase (see text). For panels (a) and (b) ${\tilde{a}}_e=0.004$ and ${\tilde{a}}_i=0.026$.Reuse & Permissions

Figure 4

The subthreshold voltage distribution corresponding to models of filtered synaptic input to layer III (circles) and layer VI (triangles) cortical neurons. The symbols correspond to a Monte Carlo simulation of a passive membrane [defined by Eq. (2)] receiving excitatory and inhibitory filtered synaptic input of the form given in Eq. (27). The lines correspond to the Gaussian, or effective-time-constant approximation defined by the moments given in Eqs. (30, 33). The passive membrane parameters are defined in the text (with $C=1\mu \mathrm{F}∕{\text{cm}}^2$) and parameters of the synaptic drive for the two models were taken from Ref. 3: the layer III neuron $\{g_{e0}=0.0295,g_{i0}=0.217,{\sigma }_e=0.00935,{\sigma }_i=0.034,{\tau }_e=7.8,{\tau }_i=8.8\}$ and the layer VI neuron $\{g_{e0}=0.0346,g_{i0}=0.165,{\sigma }_e=0.00866,{\sigma }_i=0.0191,{\tau }_e=2.7,{\tau }_i=10.5\}$. The conductances are in units of ${\text{mS}∕\text{cm}}^2$ and the time constants in milliseconds.Reuse & Permissions