Perceptron learning rule derived from spike-frequency adaptation and spike-time-dependent plasticity

Abstract

It is widely believed that sensory and motor processing in the brain is based on simple computational primitives rooted in cellular and synaptic physiology. However, many gaps remain in our understanding of the connections between neural computations and biophysical properties of neurons. Here, we show that synaptic spike-time-dependent plasticity (STDP) combined with spike-frequency adaptation (SFA) in a single neuron together approximate the well-known perceptron learning rule. Our calculations and integrate-and-fire simulations reveal that delayed inputs to a neuron endowed with STDP and SFA precisely instruct neural responses to earlier arriving inputs. We demonstrate this mechanism on a developmental example of auditory map formation guided by visual inputs, as observed in the external nucleus of the inferior colliculus (ICX) of barn owls. The interplay of SFA and STDP in model ICX neurons precisely transfers the tuning curve from the visual modality onto the auditory modality, demonstrating a useful computation for multimodal and sensory-guided processing.

Fig. 1.

A single-neuron model for instructive coding. (A) Neuron receives auditory input () of short latency and visual input (v) of longer latency. (B) Auditory-input synapse is subject to STDP, i.e., it strengthens when the neuron fires action potentials after presynaptic spikes (positive time difference) and weakens in the contrary case. The STDP pairing function shown has exponential tails. (C) Conductance-based integrate-and-fire neuron exhibits SFA, illustrated by its response to a 50 ms step input. The spike rate in response to the onset of the step input is high but then quickly adapts.

Fig. 2.

Under SFA, visual responses, V, to a multimodal stimulus report the alignment of preceding auditory responses, A. (A) (Top) Auditory and visual input currents, I_A and I_V (solid lines), are approximated as step functions of duration, T (dashed lines). T_lat, visual latency. (Middle) Firing rate, R(t), adapts during the auditory input and leads to reduced firing during the equally strong visual input. U(t), membrane potential. (Bottom) Fast buildup and slow decay of the adaptation conductance, g_K. (B) Visual response, V, is a linear decreasing function of the auditory response, A. The different curves correspond to visual input current (I_V), varying from 0.95 to 6.35 nA in 1.35 nA increments along the direction of the arrow. This analytical relationship between A and V (Eq. 1) is linear for large I_V values. The linear relationship is also a good approximation outside this bound (the bound of the linear range in Eq. 2 is indicated by the dotted line, and the extrapolation of the linear relationships is indicated by the dashed lines).

Fig. 3.

The combination of STDP and SFA implements the delta learning rule. (A) When afferent synapses are weak, auditory inputs do not elicit auditory responses. As a result, the neuron [with membrane potential U(t)] responds strongly to the visual input, thus leading to strengthening of the synaptic conductance, g^A, of a representative auditory synapse. (B) When auditory afferent synapses are strong, the neuron’s firing adaptation to auditory inputs results in more post-pre spike pairings, and thus leads to depression of synaptic strength. (C) Spiking-neuron simulation result of vector field illustrating the interplay between synaptic weight changes and auditory–visual responses. The lengths of the arrows are proportional to the absolute value of Δg (arbitrarily scaled units), and their directions indicate the effects Δg has on auditory responses, A, and visual responses, V. For example, an arrow pointing to the left indicates a negative weight change (depression) with the effect of reducing A, and an arrow pointing to the lower right indicates a positive weight change (potentiation) that increases A and decreases V. Contour lines of Δg (solid lines, in increments of 5 × 10⁻¹⁶ S) are roughly parallel to bV − A, where b = 0.143 is a constant. Along the contour line, 0, potentiation balances depression (equilibrium point). This plot was produced for the fixed presynaptic firing rate = 250 Hz and looked qualitatively similar for values of in the range of 50–350 Hz. (D) Contour lines as in C but based on numerical evaluation of Eq. 4. The bound (Eq. 2) for which the delta rule (Eq. 5) is exact is indicated by the dotted line. For small V, the extrapolation of the linear relationships in Eq. 5 (dashed lines) is a good approximation of the true nonlinear behavior. (E) Synaptic weight changes, Δg, depend linearly on the presynaptic firing rate (in Hz) for different values of V − A (spiking-neuron simulations).

Fig. 4.

Map formation in the ICX. (A) We model a single ICX neuron receiving auditory inputs from pools of ICC neurons and visual inputs from a pool of OT neurons. The auditory and visual tuning functions, (θ) and υ(θ), respectively, are bell-shaped. (B) Equilibrium ICX auditory response tuning, A(θ), (dashed line) is roughly proportional to OT tuning curves, υ(θ), (full line, with a suitable scaling factor α), thereby qualifying OT inputs as perceptron-like teacher signals.