Neuronal Temporal Filters as Normal Mode Extractors

TL;DR

This work addresses how neurons might predict future inputs under physiological delays by framing prediction as a local linearization problem and deriving Neuronal Temporal Filters through Normal Mode Decomposition (NMD). A neuron learns the top left eigenvector of a generalized eigenproblem and projects its lag-embedded input onto the corresponding subspace to extract the fastest-growing dynamical mode, effectively predicting future trends. The study shows that the resulting temporal filter’s shape transitions from monophasic to multiphasic as the input SNR increases, aligning with observed SNR-dependent STA shapes in biology, and provides analytical and numerical analyses of noise effects and comparisons to an optimal projection approach. The findings offer a dynamics-based, mechanistic account of predictive neural computation and propose plausible neural implementations, including stacking layers to uncover higher-order latent structure. Overall, the work links normal-mode dynamics, eigenvector-based projection, and predictive coding concepts to explain how single neurons could anticipate future inputs.

Abstract

To generate actions in the face of physiological delays, the brain must predict the future. Here we explore how prediction may lie at the core of brain function by considering a neuron predicting the future of a scalar time series input. Assuming that the dynamics of the lag vector (a vector composed of several consecutive elements of the time series) are locally linear, Normal Mode Decomposition decomposes the dynamics into independently evolving (eigen-)modes allowing for straightforward prediction. We propose that a neuron learns the top mode and projects its input onto the associated subspace. Under this interpretation, the temporal filter of a neuron corresponds to the left eigenvector of a generalized eigenvalue problem. We mathematically analyze the operation of such an algorithm on noisy observations of synthetic data generated by a linear system. Interestingly, the shape of the temporal filter varies with the signal-to-noise ratio (SNR): a noisy input yields a monophasic filter and a growing SNR leads to multiphasic filters with progressively greater number of phases. Such variation in the temporal filter with input SNR resembles that observed experimentally in biological neurons.

Figures (7)

• Figure 1: Neuronal temporal filters. (A,B) Experimentally obtained by Spike Triggered Average (STA) from two retinal ganglion cells stimulated by white noise of different contrast. (C) Top left eigenvectors obtained in our framework for different levels of noise in a synthetic example. Shading shows standard deviation. Whereas theoretical filters peak when the neuron spikes, experimental ones peak before that. This artifact is due to signal processing delays from photo-absorption to the ganglion cell firing and the smoothness of experimental filters resulting from the low-pass nature of biophysical processes.
• Figure 2: Problem formulation. (A) We generate a number of trajectories in the vicinity of a hyperbolic fixed point. Each trajectory is comprised of a number of growing and decaying exponentials and white noise (see panel C). (B) From these trajectories, we learn the top left eigenvector of the inferred ${\bf A}$ matrix. We use this as the time kernel of our proposed neuron. (C) We apply the time kernel to a previously unseen trajectory from the same dynamical system. (D) We extract the most dominant mode by convolving the time series with the computed time kernel. In this example, the presence of the dominant mode is clear from the projection starting at $t=1$. However, if we only look at the time series in (C), the presence of a growing mode would not be clear until around time $t=2$. Dynamics details: the time series is comprised of five different exponents with exponents given by $\{-1.5, -1, 0, 1, 1.5\}$. The time series is discretized with time-step 0.05.
• Figure 3: Analytically derived eigenvectors and eigenvalues for the lag $n=5$ for different noise levels. The $x$ axis for the eigenvalue plots denotes the eigenvalues sorted from largest to smallest real part. System with exponents $\{0.7, 0.2, -0.1, -0.4, -1.6\}$ and time-step equal to 1 for simplicity.
• Figure 4: Dependence of the filter shape for the rank 5 system in Fig. \ref{['fig:sample_series']} on system parameters: (A) noise amplitude, (B) system order (the number of exponential constituents, $k$), and (C) length of the lag vector, $n$. In (B) and (C) the noise is fixed at $0.01\%$. In (B), to achieve a comparable system of lower rank $k$, we keep the top $k$ exponents of the system.
• Figure 5: Comparison of the performance with respect to the mean square error between the recovered dominant exponential and the ground truth. The 'noise optimal' solution is given by the filter in Eq. \ref{['eq:noise_optimal']} and 'noiseless filter' refers to using the filter compted in the absence of noise but applied to noisy observations. The system is the same as in Fig. \ref{['fig:sample_series']}.