Constructing Dampened LTI Systems Generating Polynomial Bases

We present an alternative derivation of the LTI system underlying the Legendre Delay Network (LDN). To this end, we first construct an LTI system that generates the Legendre polynomials. We then dampen the system by approximating a windowed impulse response, using what we call a "delay re-encoder". The resulting LTI system is equivalent to the LDN system. This technique can be applied to arbitrary polynomial bases, although there typically is no closed-form equation that describes the state-transition matrix.

Discrete Function Bases and Convolutional Neural Networks

We discuss the notion of "discrete function bases" with a particular focus on the discrete basis derived from the Legendre Delay Network (LDN). We characterize the performance of these bases in a delay computation task, and as fixed temporal convolutions in neural networks. Networks using fixed temporal convolutions are conceptually simple and yield state-of-the-art results in tasks such as psMNIST. Main Results (1) We present a numerically stable algorithm for constructing a matrix of DLOPs L in O(qN) (2) The Legendre Delay Network (LDN) can be used to form a discrete function basis with a basis transformation matrix H in O(qN). (3) If q < 300, convolving with the LDN basis online has a lower run-time complexity than convolving with arbitrary FIR filters. (4) Sliding window transformations exist for some bases (Haar, cosine, Fourier) and require O(q) operations per sample and O(N) memory. (5) LTI systems similar to the LDN can be constructed for many discrete function bases; the LDN system is superior in terms of having a finite impulse response. (6) We compare discrete function bases by linearly decoding delays from signals represented with respect to these bases. Results are depicted in Figure 20. Overall, decoding errors are similar. The LDN basis has the highest and the Fourier and cosine bases have the smallest errors. (7) The Fourier and cosine bases feature a uniform decoding error for all delays. These bases should be used if the signal can be represented well in the Fourier domain. (8) Neural network experiments suggest that fixed temporal convolutions can outperform learned convolutions. The basis choice is not critical; we roughly observe the same performance trends as in the delay task. (9) The LDN is the right choice for small q, if the O(q) Euler update is feasible, and if the low O(q) memory requirement is of importance.

Passive nonlinear dendritic interactions as a general computational resource in functional spiking neural networks

Nonlinear interactions in the dendritic tree play a key role in neural computation. Nevertheless, modeling frameworks aimed at the construction of large-scale, functional spiking neural networks tend to assume linear, current-based superposition of post-synaptic currents. We extend the theory underlying the Neural Engineering Framework to systematically exploit nonlinear interactions between the local membrane potential and conductance-based synaptic channels as a computational resource. In particular, we demonstrate that even a single passive distal dendritic compartment with AMPA and GABA-A synapses connected to a leaky integrate-and-fire neuron supports the computation of a wide variety of multivariate, bandlimited functions, including the Euclidean norm, controlled shunting, and non-negative multiplication. Our results demonstrate that, for certain operations, the accuracy of dendritic computation is on a par with or even surpasses the accuracy of an additional layer of neurons in the network. These findings allow modelers to construct large-scale models of neurobiological systems that closer approximate network topologies and computational resources available in biology. Our results may inform neuromorphic hardware design and could lead to a better utilization of resources on existing neuromorphic hardware platforms.

Point Neurons with Conductance-Based Synapses in the Neural Engineering Framework

The mathematical model underlying the Neural Engineering Framework (NEF) expresses neuronal input as a linear combination of synaptic currents. However, in biology, synapses are not perfect current sources and are thus nonlinear. Detailed synapse models are based on channel conductances instead of currents, which require independent handling of excitatory and inhibitory synapses. This, in particular, significantly affects the influence of inhibitory signals on the neuronal dynamics. In this technical report we first summarize the relevant portions of the NEF and conductance-based synapse models. We then discuss a na\"ive translation between populations of LIF neurons with current- and conductance-based synapses based on an estimation of an average membrane potential. Experiments show that this simple approach works relatively well for feed-forward communication channels, yet performance degrades for NEF networks describing more complex dynamics, such as integration.