Membrane potential and spike train statistics depend distinctly on input statistics

Robert Rosenbaum and Krešimir Josić

Phys. Rev. E 84, 051902 – Published 4 November 2011

Abstract

A description of how the activity of a population of neurons reflects the structure of its inputs is essential for understanding neural coding. Many studies have examined how inputs determine spiking statistics, while comparatively little is known about membrane potentials. We examine how membrane potential statistics are related to input and spiking statistics. Surprisingly, firing rates and membrane potentials are sensitive to input current modulations in distinct regimes. Additionally, the correlation between the membrane potentials of two uncoupled cells and the correlation between their spike trains reflect input correlations in distinct regimes. Our predictions are experimentally testable, provide insight into the filtering properties of neurons, and indicate that care needs to be taken when interpreting neuronal recordings that reflect a combination of subthreshold and spiking activity.

Received 3 August 2011

DOI:https://doi.org/10.1103/PhysRevE.84.051902

Authors & Affiliations

Robert Rosenbaum¹ and Krešimir Josić^1,2

¹Department of Mathematics, University of Houston, Houston Texas 77204-3008, USA
²Department of Biology and Biochemistry, University of Houston, Houston Texas 77204-5001, USA

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Vol. 84, Iss. 5 — November 2011

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Images

Figure 1
(a) Firing rate ( $r_{s}$ , dashed red line, top), mean membrane potential ( $〈 V 〉$ , solid blue line, top) and gains ( $d r_{s} / d μ$ , dashed red line, bottom; $d 〈 V 〉 / d μ$ , solid blue line, bottom) as functions of the excitatory input rate $r_{e}$ . (b) Susceptibility magnitude of the firing rate. (c) Susceptibility magnitude of the mean membrane potential. As the level of excitation increases, firing rates become more sensitive and membrane potentials become less sensitive to perturbations. In all plots $r_{i} = 2$ KHz and $τ_{m} = 20$ ms. Voltage is scaled so that $V_{re} = 0$ and $V_{th} = 1$ with $J_{e} = J_{i} = 1 / 30$ . Mean membrane potential has units ${(V_{th} - V_{re})}^{- 1}$ , $ω$ has units Hz, and susceptibility functions have units ${(V_{th} - V_{re})}^{- 1}$ for $χ_{s}$ and ms for $χ_{V}$ .Reuse & Permissions
Figure 2
(a) Cross covariance between spike trains as $r_{e}$ increases. Inset compares linear response calculation (solid) to the strong excitation limit [dashed line, from Eq. (8)] when $r_{e} = 4.5$ KHz. (b) Cross covariance between membrane potentials as $r_{e}$ increases. Inset compares linear response calculation (solid) to the weak excitation limit [dashed line, from Eq. (4)] when $r_{e} = 2.15$ KHz. Parameters are the same as in Fig. 1 with input cross covariances $C_{e} (τ) = ρ_{in} r_{e} e^{- | τ | / τ_{in}} / τ_{in}$ , $C_{i} (τ) = ρ_{in} r_{i} e^{- | τ | / τ_{in}} / τ_{in}$ , and $C_{e i} (τ) = 0$ so that, from Eq. (2), $C_{in} (τ) = ρ_{in} D e^{- | τ | / τ_{in}} / τ_{in}$ with input correlation magnitude $ρ_{in} = 0.1$ and time scale $τ_{in} = 5$ ms. Axes have units ms for $τ$ , KHz for $r_{e}$ , Hz $^{2}$ for $C_{s} (τ)$ , and ${(V_{th} - V_{re})}^{2}$ for $C_{V} (τ)$ . Firing rates vary in range from $0.1$ to 58 Hz.Reuse & Permissions
Figure 3
(a) Spike count correlations and (b) normalized membrane potential cross correlation at various firing rates (see inset). Linear response approximations (solid line) are compared to simulations with Poisson inputs (dashed line). Firing rates were modulated by changing $r_{e}$ . All other parameters are as in Fig. 2.Reuse & Permissions
Figure 4
(a) Spike count correlation over large time windows and (b) peak membrane potential correlation plotted against firing rate as $r_{e}$ and $r_{i}$ vary along linear paths: $r_{e} - 500 Hz = α (r_{i} - 500 Hz)$ for different slopes $α$ (see inset). All other parameters are as in Fig. 2.Reuse & Permissions
Figure 5
Spike count correlation over large time windows versus peak membrane potential correlation for 500 randomly generated excitatory and inhibitory input rates and correlations. (a) Input rates were drawn from wide uniform distributions ( $r_{e} \in [2, 4]$ KHz and $r_{i} \in [0, 1.75]$ KHz) and input correlations from narrow uniform distributions ( $ρ_{e e} \in [0.15, 0.2]$ , $ρ_{i i} \in [0.15, 0.2]$ , and $ρ_{e i} = 0$ ). (b) Input rates were drawn from narrower uniform distributions ( $r_{e} \in [2.2, 2.4]$ KHz and $r_{i} \in [1.3, 1.4]$ KHz) and input correlations from wider uniform distributions ( $ρ_{e e} \in [0, 0.2]$ , $ρ_{i i} \in [0, 0.2]$ , and $ρ_{e i} = 0$ ). Parameters are the same as in Fig. 2 except input cross covariances are $C_{e} (τ) = ρ_{e e} r_{e} e^{- | τ | / τ_{in}} / τ_{in}$ , $C_{i} (τ) = ρ_{i i} r_{i} e^{- | τ | / τ_{in}} / τ_{in}$ , and $C_{e i} (τ) = 0$ .Reuse & Permissions
Figure 6
(a) Mean membrane potential and an approximation to the gain (found by taking $Δ 〈 V 〉 / Δ μ$ for the points sampled) for the spiking EIF model as a function of $r_{e}$ . Dots show sampled points, which are interpolated linearly. (b) Membrane potential cross correlation for a spiking EIF model plotted for various values of $r_{e}$ . (c) A sample voltage trace (taken at $r_{e} = 2.7$ KHz) and the trajectory of a single spike for the spiking EIF. (d) Cross correlations between two pooled recordings of 200 membrane potential traces, obtained by applying Eq. (17) to the cross correlations in (b) with $n = 200$ . Note that correlations are at least an order of magnitude larger here than in (b) due to pooling. Input parameters are as in Fig. 2.Reuse & Permissions

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