Fig. 5: Inductive bias of adLIF gradients.
From: Advancing spatio-temporal processing through adaptation in spiking neural networks
a Computational graph showing the state-to-state derivative \(\frac{\partial s[t]}{\partial s[t-1]}\) back-propagating through time. s[k] denotes the state vector (Eq. (5)). b Response of the membrane potential of a LIF neuron (left) and an adLIF neuron (right) to a single input spike. The shape of the derivative \(\frac{\partial u[T]}{\partial u[t]}\) (bottom) matches the reversed impulse response function. c Comparison of the derivative \(\frac{\partial {{{\mathcal{L}}}}}{\partial \theta }\) for a wavelet input current. The multiplication from Eq. (12) of the input current with the state derivative is schematically illustrated for both, the adLIF and the LIF case. The frequency of the wavelet approximately matches the intrinsic frequency of the oscillation of the membrane potential oscillation of the adLIF neuron. The bar plot on the bottom shows the derivative \(\frac{\partial {{{\mathcal{L}}}}}{\partial \theta }\) for both neurons, where color indicates input amplitude. d Same as panel c but for a constant input current. e Same as panel c but for different positions of the wavelet current. Middle plot shows the alignment between input and back-propagating derivative \(\frac{\partial u[T]}{\partial u[t]}\) for the adLIF neuron. Input wavelet is given with a phase shift of 0,\(-\frac{1}{2}P\) and \(-\frac{3}{4}P\) with respect to the period P of the adLIF neuron oscillation.
