A Hebbian learning rule gives rise to mirror neurons and links them to control theoretic inverse models

Abstract

Mirror neurons are neurons whose responses to the observation of a motor act resemble responses measured during production of that act. Computationally, mirror neurons have been viewed as evidence for the existence of internal inverse models. Such models, rooted within control theory, map-desired sensory targets onto the motor commands required to generate those targets. To jointly explore both the formation of mirrored responses and their functional contribution to inverse models, we develop a correlation-based theory of interactions between a sensory and a motor area. We show that a simple eligibility-weighted Hebbian learning rule, operating within a sensorimotor loop during motor explorations and stabilized by heterosynaptic competition, naturally gives rise to mirror neurons as well as control theoretic inverse models encoded in the synaptic weights from sensory to motor neurons. Crucially, we find that the correlational structure or stereotypy of the neural code underlying motor explorations determines the nature of the learned inverse model: random motor codes lead to causal inverses that map sensory activity patterns to their motor causes; such inverses are maximally useful, by allowing the imitation of arbitrary sensory target sequences. By contrast, stereotyped motor codes lead to less useful predictive inverses that map sensory activity to future motor actions. Our theory generalizes previous work on inverse models by showing that such models can be learned in a simple Hebbian framework without the need for error signals or backpropagation, and it makes new conceptual connections between the causal nature of inverse models, the statistical structure of motor variability, and the time-lag between sensory and motor responses of mirror neurons. Applied to bird song learning, our theory can account for puzzling aspects of the song system, including necessity of sensorimotor gating and selectivity of auditory responses to bird's own song (BOS) stimuli.

Keywords: inverse problem; linear models; mirror neurons; sensory motor learning; songbird.

Figure 1

Spiking activity in single neurons of singing zebra finches, illustrating (A) a variable premotor code in LMAN and (B) a stereotyped code in HVC. (A) Spike raster plot of LMAN projection neuron aligned to 41 renditions of the stereotyped song motif. Three exemplary sound oscillograms of the motif are shown on top. The neuron produces single spikes and spike bursts at different times in each rendition of the motif. The motif-averaged firing rate is shown at the bottom. (B) Spike raster plot of HVC projection neuron aligned to 22 renditions of the stereotyped song motif (in a different bird), three exemplary motif oscillograms are shown on top. In each rendition of the motif the neuron produces a brief burst of spikes at precisely the same time.

Figure 2

Delayed feedback and inverse model, illustrated by vocal production in birds. In our model of delayed sensory feedback the auditory response a(t) in a sensory area at time t depends linearly on motor activity m(t − τ) in a motor brain area at an earlier time t-τ according to a(t) = Qm(t − τ), where Q is the unknown motor-sensory mapping and τ the unknown delay of auditory feedback. An inverse V is a mapping from sensory neurons back onto motor neurons that inverts the action of Q: V = Q⁻¹.

Figure 3

Cross-correlation functions for variable and stereotyped motor codes. (A) In a variable motor code m(t) (shaded area). Activity bursts m₁ (black) and m₂ (blue) of width t₀ in two example motor neurons occur at diverse time lags relative to each other across renditions of the song motif. Auditory tuning in the shown sensory neuron is such that it responds a₁ to bursts m₁ after a time lag τ. Repeated co-activation m₁ → a₁ and non-zero eligibility e(τ) (red bar) at time lag τ leads to increased synaptic weight V₁₁ (red arrow) and to a causal inverse. Lack of correlation between m₂ and a₁, as well as heterosynaptic competition, prevents V₂₁ from similarly increasing (blue thin arrow). (B) The cross-correlation function C_ij(t) for variable codes is flat except the auto-correlation peak at zero time lag (motor activity is uncorrelated among neuron pairs). Note: based on square activity pulses in motor neurons in (A) the true cross-correlation shape is triangular (blue dotted line) which we approximate by a square pulse of width t₀ ≃ 10 ms. The auto-correlation peak height is C₀. (C) In a stereotyped motor code m(t) (shaded area), bursts m₁ (black) and m₂ (blue) occur at a fixed time lag relative to each other across renditions of the song motif (traveling pulse of activity). Repeated co-activation m₂ → a₁ at higher eligibility (red bar) than the eligibility of m₁ → a₁ leads to strengthening of synapse V₂₁ (red arrow) and to a predictive inverse. (D) The cross-correlation function C_ij(t) for stereotyped codes peaks also at non-zero time lags.

Figure 4

The mirroring offset is determined in recordings of neural activity across (A) singing and (B) auditory stimulation by song playback; the offset (red) is large for causal inverses (middle column) and close to zero for predictive inverses (right column). (A) During singing, a motor-neuron burst m₁ drives a song feature (green notes) after a time delay τ_m. (B) Playback of that song feature leads to auditory response a₁ after a time delay τ_a. And, auditory response a₁ leads to motor neuron response m^a₁ in case of a causal inverse (middle column) and to motor neuron response m^a₂ in case of a predictive inverse (right column) after an additional time lag τ_s (spike propagation time from auditory to motor area) that is assumed to be 0 for simplicity. Thus, after alignment of motor activity and sensory response with song (green arrows), in the case of a causal inverse, the mirroring offset Δt defined as the time lag between motor activity and playback response (red bar with extension set by red dashed lines) is equal to the sensory feedback delay τ = τ_m + τ_a, whereas in the case of a predictive inverse the mirroring offset Δt is close to 0. The reason for the 0 offset associated with a predictive inverse is that the auditory burst a₁ driving the playback response m^a₂ is selective to the sound feature that during singing was generated by the much earlier burst m₁ in a different neuron (black burst in (A), right panel), but not the feature generated by m₂ (blue burst in (A), right panel).