Impact of redundancy on stable decoding

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Neural activity is redundant: many states in motor cortex can generate similar movements. When we record from motor cortex, we capture only a small fraction of the total neurons. Redundancy makes it possible to observe the overall state of motor cortex from limited observations, but might also impair the generalization performance of a linear decoder.

Consider two neurons, $A$ and $B$, that combine linearly to produce movement $C={\alpha }_{1}A+{\alpha }_{2}B$. (Perhaps both neurons drive the same targets in spinal cord.) An animal could use any linear combination of activations of units $A$ and $B$ to perform behavior $C$, so long as the sum ${\alpha }_1+{\alpha }_2$ is constant. What if there is an unobserved variable $\gamma$ that sets whether neuron $A$ or $B$ is used more (Fig. 1)?

Figure 1: (simulated hypothetical scenario) Neural signals $A$ and $B$ combine linearly according to weight $\gamma$ to form behavioral output $C=\gamma A+(1-\gamma )B$. Parameter $\gamma$ modulates sinusoidally between $0.25$ and $0.75$.

Let's say we record only from neuron $A$. Building a linear decoder $\hat{C}=\alpha A$  leads to an over-fit (and erroneous) estimate of the contribution of $A$ to behavior: $\hat{\alpha }=(A\cdot C)/(A\cdot A)\approx 0.996$. When predicting behavior from $A$, the reconstruction error varies depending on the unobserved slow variable $\gamma$ (figure 2). This error resembles transient noise, or perhaps an independent source of neuronal variability. But, the activation of $A$ and $B$ always drives behavior in a predictable way. Hidden sources of variability, and under-sampling of the neural population, leads to apparent instability when there is none (Fig. 2).

Figure 2: (simulated hypothetical scenario) (A) Reconstructed behavior using only unit $A$ leads to unstable decoding accuracy. (B) The the smoothed (Gaussian kernel σ=60 ms) absolute reconstruction error varies with this hidden parameter $\gamma$, which sets $A$'s contribution to the motor output.