Fig. 3: Workflow to calculate ε-discriminable inputs.

In the first step (a–c), we obtain the distribution P(a_T∣h) of pure output activity in a reservoir (with parameter λ) subject to an input h. a For T → ∞, these distributions become δ-distributions that we calculate using our mean-field approximation. b For finite T, we perform many numerical simulations and fit a Beta distribution to the data using maximum likelihood estimates (orange examples obtained for T = 100). We then use these fit results to train a deep neural network as a general function approximation that interpolates the fit parameters (α, β) for all simulation parameters (λ, h, T). c In the limit T → 0, we obtain analytical results by solving the Fokker–Planck equation that are in good agreement with corresponding numerical data for T = 1 (yellow histograms). d In the next step, we obtain the distribution of noisy output responses P(o^T∣h) by a convolution of P(a^T∣h) with a Gaussian \({{\mathcal{N}}}(0,{\sigma }^{2})\) of small variance σ². This step allows (i) to connect to previous mean-field results for T → ∞¹¹ and (ii) circumvents numerical intricacies for finite N at the boundaries. The example compares beta distributions from the neural network interpolations (orange) with their corresponding noisy distributions (gray), which mostly differ at the boundaries. e, f In the last step, we determine two sets of discriminable inputs that can be discriminated from reference distributions for vanishing input (left Gaussian distribution, black) and diverging input (right Gaussian distribution, black). For this, we start from the left and right references and perform iterative bisection searches in h to find input values whose response distributions overlap exactly ε with the previous one. The dynamic range is calculated from the smallest and largest inputs (marked green). The number of discriminable inputs is obtained as the average size of the sets. Examples are shown for λ = 0.999, ε = 0.1, σ = 0.01. Example distributions for \(h\in \left[5.6\cdot 1{0}^{-5},1.8\cdot 1{0}^{-3},5.6\cdot 1{0}^{-3},1.8\cdot 1{0}^{-2},3.2\cdot 1{0}^{-1}\right]\).