Digital.CSIC. Repositorio Institucional del CSIC

oai:digital.csic.es:10261/380653

S1 Appendix. Full derivation of the mean phase of firing.Provides a detailed solution to the deterministic part of Eq 1, resulting in the “rate-to-phase” transfer function (Eq 3) previously derived in [6]., S2 Appendix. Full derivation of the variance of phase of firing. Explains the use of a first-order Taylor series expansion and the propagation of uncertainty around the spike threshold to derive an analytical approximation of the phase variance., S3 Appendix. Approximation of information rate. Describes the approximation of the entropy of Gaussian mixtures to derive an analytical estimation of the information rate. Additionally, it introduces a correction factor to account for cycle-to-cycle correlations., S4 Appendix. Neuron parameters. Includes: Table A. Default parameters for hippocampal neurons; Table B. Neuron parameters along the hippocampal dorsoventral axis; Table C. Neuron parameters for visual and olfactory cells. -- S5 Appendix. Simulations. Details the numerical integration of Eq 1 used in simulations supporting our theoretical framework., S1 Fig. Oscillatory frequency modulates the tonic input range that makes neurons phase-lock. (A) Phase-locking function for different values of the oscillatory frequency f (color coded). Shaded areas denote the phase-locking range of Is, corresponding to the domain of Is in Eq 3. The parameters are the same ones used in [6] and in Fig 1, to match hippocampal physiology (described in Table A in Appendix). Note that the phase-locking range spans half of a cycle and not the full cycle as previously thought, in agreement with recent studies [7]. (B) Length of Is range (max(Is) − min(Is) = 2AIosc) across frequencies. (C) Average Is as (middle point in Is range) across frequencies. (D) Effective amplitude of the membrane potential oscillation, Vosc produced by the oscillatory input Iosc. Given that the membrane acts as a low-pass filter, determined by τm, the effective oscillation in the membrane potential can be found to be Vosc = RmIoscA. Thus, since the membrane filters the oscillatory input as , the amplitude of the membrane potential Vosc will decrease with f approximately as ∼ 1/f., S2 Fig. Analytical approximation of phase distributions. (A) Phase distributions for a range of frequencies and noise strengths. Histograms denote the simulations whereas solid lines denote the theoretical predictions. For the simulations, first-spike phases are recorded from the beginning of the second cycle (with the trough as ϕ = 0), after initializing the neurons to their expected phase ϕ0 = μϕ to allow them to reach steady-state dynamics (as described in Appendix). The parameters used here are described in Table A in Appendix). (B) Average variance in rad2 (across Is levels) across a wide frequency–noise parameter space, for simulations and the theoretical predictions. (C) Diagonal slices of plots in (B), showing the deviation of the theory from the simulations after a certain level of noise amplitudes at high frequencies, due to the bounded variance of simulated spike phases constrained to the measurable range of [0, 2π] radians., S3 Fig. Effective rhythmic input sampling. (A) An example signal with τs of 100 ms sampled by different oscillation frequencies. (B) Effective frequency for various τs values., S4 Fig. Information rate across frequencies for the range of physiologically realistic noise levels (η)., S5 Fig. Normalized information rate across the frequency–noise parameter space for simulations and theoretical predictions., S6 fig. Normalized information rate across the frequency–noise parameter space for a wide range of input signal time constants τs., S7 Fig. Normalized information rate across the frequency–noise parameter space for a wide range of membrane time constants τm., S8 Fig. Optimal frequency for a wide range of membrane time constants τm and input signal time constants τs. At every point of the τm − τs parameter space (logarithmically discretized in a 200 × 200 grid), we computed rnorm over the frequency–noise space (as in e.g., Fig 4B). Then, the optimal frequency was estimated as an average of the peak frequency between the physiologically-realistic noise range η = [0.1, 0.15]., S9 Fig. Normalized information rate for colored noise with different long-range correlation lengths: White, pink, and brown noise.A value of 100 ms was used here for τs. All plots represent the results of simulations., S10 Fig. Normalized information rate across the dorsoventral axis for simulations and theoretical predictions., S11 Fig. Normalized information rate across the frequency-amplitude space for simulations and theoretical predictions., Peer reviewed