Fig 1.
Basic Principles of the Proposed Model.
a) Random sampling with interspike intervals (ISIs): Each spike in a train is interpreted as carrying an analog label, whose value corresponds to the length of the ISI preceding the spike, i.e. to the difference in spike times between the considered spike and its predecessor (see numbers above spikes in arbitrary units). These analog values are therefore samples from the ISI distribution p(t) underlying the spike train. Depending on the computational context, p(t) may be stationary or not. (Figure adapted with permission from [25]) b) EIF neuron as ISI sampler and probability transducer: A ‘user-defined’, target ISI distribution pin(t) that is given by a small, multiplicative modulation (left black trace) of an exponential distribution (blue/red dotted trace) is specified as input current to the EIF neuron. Subject to noise, the neuron responds with a membrane potential fluctuating around some baseline value and thus with the stochastic firing of output spikes, whose ISI distribution also follows a modulated exponential distribution pEIF(t) (right black trace). Our theory shows that modulations at the output may approximate those at the input, provided the latter stay within sufficiently small margins (blue/red shaded areas). Approximations are improved upon increasing the baseline level of the membrane potential towards firing threshold.
Table 1.
Parameters underlying the numerical simulations of the EIF/LIF neuron and the theoretical results of section 3.2.
Fig 2.
Dynamical Situation of the Membrane Potential During UP States of the EIF Neuron: Our approach assumes the membrane potential V(t) = V0 + ΔV(t) (blue trace) to fluctuate around some constant value V0, with time-dependent fluctuations ΔV(t).
V0 is supposed to be close to the firing threshold VT (and generally far from the resting (leak reversal) potential EL), such that the exponential input current of the EIF (red trace) becomes effective. The exponential term is crucial for an approximation of the ideal ISI sampler (see main text for details).
Fig 3.
Bode Plots of the EIF and LIF Neuron for Various Different Baseline Voltages V0 in the High Conductance Regime (gL = 5·30nS, K ≈ 53.4).
Top row: Phase shift of TEIF(s) (left) and TLIF(s) (right) as a function of frequency, for baseline voltages V0 ∈ [−65mV, VT − 1mV] (colors). For the EIF neuron, the curve for V0 = VT − 1mV is drawn as dashed, black line and replotted on the right for comparison. Bottom row: Amplitude gain ∣TEIF(s)∣ (left) and ∣TLIF(s)∣ (right) for baseline voltages V0 ∈ [−65mV, …, VT − 1mV]. For the EIF neuron, the curve for V0 = VT − 1mV is drawn as dashed, black line and replotted in the right plot for comparison.
Fig 4.
Bode Plots of the EIF and LIF Neuron for Various Different Baseline Voltages V0 in the Low Conductance Regime (gL = 30nS, K ≈ 10.7). See Fig 3 for legend.
Fig 5.
EIF/LIF Performance of Approximating a Predefined ISI Distribution.
Top: Log-modulation functions ln Δpin(t), ln ΔpEIF(t) and lnΔpLIF(t). Red trace gives the ideal, target log-modulation function ln Δpin(t) (broadband signal consisting of a superposition of 60 sinusoids with random phase, unit amplitude and frequencies taken equidistantly from [10, 200]Hz. Baseline voltage was set to V0 = −51.4mV, corresponding to a baseline hazard of h0 = 7.2Hz. Probability fluctuation ratio was set to rΔp = 0.44). Blue and orange traces are the actual log-modulation functions ln ΔpEIF(t) and lnΔpLIF(t) of the EIF and LIF neuron respectively, obtained by numerical integration (see methods). Bottom: ISI distributions pin(t), pEIF(t) and pLIF(t) corresponding to the modulation functions in the upper plot.
Fig 6.
Summary Results of Sweeping V0 and rΔp for Low-Frequency Log-Probability Modulation.
a) EIF neuron: Shown is a color-coded contour plot of L1norm(ln Δpin, ln ΔpEIF) for various combinations of the probability fluctuation ratio rΔp (x-axis) and the baseline voltage V0 (y-axis). The curves are example log-probability modulation functions ln Δpin (red) and ln ΔpEIF (blue). The modulations lnΔpLIF of the LIF neuron are also shown for comparison (orange). Values of rΔp and V0 of each example are indicated by blue arrows. b) LIF neuron: Color-coded surface plot of L1norm(ln Δpin, lnΔpLIF), color code as in (a).
Fig 7.
Summary Results of Sweeping V0 and rΔp for High-Frequency Log-Probability Modulation.
The same plots as in Fig 6, but for high-frequency modulation.
Fig 8.
Range of Realizable Values of p(t) as a Function of Time.
Shown are the color-coded distributions pp(t) of values p(t) for different probability fluctuation ratios rΔp (rows) and hazard baselines h0 (columns). For any fixed time t, pp(t) gives the distribution of values of the ISI distribution p(t). Blue lines give the baseline ISI distributions p0(t) = h0 exp(h0 t) that are induced in the absence of probability modulation (rΔp = 0).
Fig 9.
ISI Sequences of the AdEx Neuron Correspond to Sequences of Random Samples from a Markov Chain.
Top: Illustration of the spike-triggered dynamics of adaptation current w (Eq 4.1), when voltage fluctuations ΔV(t) can be neglected. Each spike, sequenced by i, leads to an immediate increase of w by some fixed amount b. Between spikes, the dynamics of w are governed by a leaky integrator. wi is the value of w immediately before the spike, i.e. before the addition of b. ISIi is the ISI label of the i-th spike. Middle: Combining ISIi and wi yields a state vector xi ≔ (ISIi, wi) that follows Markov dynamics. Shown is a Bayesian network representation of the resulting Markov chain. Input to the chain is given by an input current I(t) provided by the neuron’s presynaptic partners. Bottom: Detailed Bayesian network when xi is expanded into its components ISIi and wi. The dependencies shown as arrows are due to Eq 3.13 and the leaky integrator dynamics governing w(t) (Eq 4.1).