Neuronal integration of synaptic input in the fluctuation-driven regime

Abstract

During sensory stimulation, visual cortical neurons undergo massive synaptic bombardment. This increases their input conductance, and action potentials mainly result from membrane potential fluctuations. To understand the response properties of neurons operating in this regime, we studied a model neuron with synaptic inputs represented by transient membrane conductance changes. We show that with a simultaneous increase of excitation and inhibition, the firing rate first increases, reaches a maximum, and then decreases at higher input rates. Comodulation of excitation and inhibition, therefore, does not provide a straightforward way of controlling the neuronal firing rate, in contrast to coding mechanisms postulated previously. The synaptically induced conductance increase plays a key role in this effect: it decreases firing rate by shunting membrane potential fluctuations, and increases it by reducing the membrane time constant, allowing for faster membrane potential transients. These findings do not depend on details of the model and, hence, are relevant to cells of other cortical areas as well.

Figure 4.

Free membrane potential and firing rate of the model neuron with conductance input are non-monotonic functions of the balanced increase of excitation and inhibition. Dots depict the results of numerical simulations; gray lines correspond to analytical approximations. a, Mean free membrane potential as a function of the excitatory rate λ_e. The inhibitory input rate (not shown) was covaried in a balanced manner. b, SD of the free membrane potential for the same input rates as in a. Arrows indicate identical SDs, corresponding to input rates used for the simulations in d and e. c, Firing rate of the neuron for the same input rates as in a. The dashed line corresponds to a simple firing rate model (see Results and Eq. 15). Arrows as in c. d, e, Time course of the membrane potential for two different levels of synaptic bombardment, resulting in equal SD of free membrane potential fluctuations, but different firing rates (b, c, arrows). Dotted lines indicate the firing threshold. f CV of the interspike interval distributions for the same input rates as in a. For a-c and f, we simulated 50 × 20 sec of neural activity for each input condition. The statistics of interest were computed for each 20 sec trial and then averaged over all 50 trials. The SEM correspond to the diameter of the dots or was even smaller.

Figure 1.

Free membrane potential fluctuations and firing rate of the model neuron with current input increase monotonically with the balanced increase of excitation and inhibition. a, Membrane potential fluctuations elicited by excitatory and inhibitory synaptic inputs with a total rate of 2000 spikes per second and 434 spikes per second, respectively. Dotted line indicates the threshold for spike generation. b, Free membrane potential for the same input realization as in a. c, Mean free membrane potential (gray line represents the analytical expression; dots depict results of numerical simulations) as a function of the excitatory input rate λ_e. The inhibitory input rate (not shown) was covaried in a balanced manner. d, SD of the free membrane potential for the same input rates as in c. Again, the gray line represents the analytical expression, and dots show the results of numerical simulations. Firing rate of the model neuron (e) and CV of the interspike interval distribution (f), for the same input rates as in c. For c-f, we simulated 60 × 20 sec of neural activity for each input condition. The statistics of interest were computed for each 20 sec trial and then averaged over all 60 trials. The SEM corresponds to the diameter of the dots or was even smaller. pot., Potential; ISI, interspike interval.

Figure 2.

Amplitude and width of PSPs decrease with increasing synaptic bombardment. a, EPSP at a membrane potential of -70 mV (bottom trace) and -55 mV (middle trace). The top trace shows the average EPSP in the presence of synaptic background activity resulting in a mean membrane potential of -55 mV. The thin black line represents results of numerical simulations; the gray line depicts the analytical approximation. b, Analogous to a for the IPSP. c, PSP amplitudes as a function of the background excitatory rate λ_e. The inhibitory input rate (not shown) was covaried with λ_e such that the mean free membrane potential remained constant (-55 mV). Black lines represent numerical simulations (open circles: EPSP; solid circles: IPSP); gray lines show values for the analytically approximated PSPs. Large open and solid circles on the ordinate represent EPSP and IPSP amplitudes in the absence of synaptic background activity (a, b, middle traces). d, Analogous to c for PSP widths. e, Mean membrane conductance μ(G_tot) relative to the leak conductance G_l. The solid circle on the ordinate indicates a relative membrane conductance of 1, corresponding to the absence of synaptic bombardment. f, Effective membrane time constant τ_eff. Dots and error bars represent mean and SD estimated from numerical simulations. The gray line depicts the analytical approximation of the mean. The solid circle on the ordinate indicates the value of the membrane time constant with all synapses quiescent. ampl., Amplitude.

Figure 3.

Amplitude of fast PSPs is less vulnerable to synaptic bombardment. a, Three hypothetical EPSPs with identical amplitude and different widths in the absence of synaptic background activity, at a membrane potential of -55 mV. The synaptic conductance transients had time constants τ_e = 0.2, 2, and 20 msec, respectively. b, c, The three EPSPs for two different levels of background excitatory (λ_e) and inhibitory (λ_i) rates, each resulting in an average membrane potential of -55 mV. d, Amplitude of the three EPSPs as a function of the background excitatory rate (λ_e). The background inhibitory rate (not shown) was covaried with λ_e such that the average membrane potential remained at -55 mV. The solid circle on the ordinate represents the amplitude of the EPSPs in the absence of synaptic bombardment (a).

Figure 5.

Free membrane potential and firing rate as a function of excitatory (λ_e) and inhibitory (λ_i) input rates, for the model neuron with conductance input (left) and with current input (right). a, Mean free membrane potential of the model with conductance input. Values below -70 mV or above -50 mV are not displayed. b, Analogous to a for the model with current input. Note the different axes, compared with a. c, SD of the free membrane potential for the model with conductance input. Values are shown for the same input domain as in a. The dashed line indicates the -55 mV contour of the mean free membrane potential. d, Analogous to c for the model with current input. e, Firing rate of the neuron with conductance input. The input domain was constrained to rates >10^2.5 spikes per second. The dashed line is again the -55 mV contour; the solid line indicates input rates corresponding to constant free membrane potential fluctuations (2.8 mV SD). f, Analogous to e for the neuron with current input. The solid line represents input rate values corresponding to 4 mV SD of the free membrane potential.