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Preparatory activity and the expansive null-space

Nature Reviews Neuroscience volume 25, pages 213–236 (2024)Cite this article

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Abstract

The study of the cortical control of movement experienced a conceptual shift over recent decades, as the basic currency of understanding shifted from single-neuron tuning towards population-level factors and their dynamics. This transition was informed by a maturing understanding of recurrent networks, where mechanism is often characterized in terms of population-level factors. By estimating factors from data, experimenters could test network-inspired hypotheses. Central to such hypotheses are ‘output-null’ factors that do not directly drive motor outputs yet are essential to the overall computation. In this Review, we highlight how the hypothesis of output-null factors was motivated by the venerable observation that motor-cortex neurons are active during movement preparation, well before movement begins. We discuss how output-null factors then became similarly central to understanding neural activity during movement. We discuss how this conceptual framework provided key analysis tools, making it possible for experimenters to address long-standing questions regarding motor control. We highlight an intriguing trend: as experimental and theoretical discoveries accumulate, the range of computational roles hypothesized to be subserved by output-null factors continues to expand.

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Fig. 1: Preparatory and execution activity in empirical data and an artificial spiking network.
Fig. 2: Output-null factors are hypothesized to serve multiple purposes.
Fig. 3: Illustration of the null-space hypothesis and its predictions.
Fig. 4: Single-neuron responses, during a delayed-reach task, from motor cortex and from simulated models.
Fig. 5: The null-space allows low trajectory tangling.
Fig. 6: The ability to estimate interpretable factors allowed experiments to address key questions regarding motor cortex.
Fig. 7: Data and simulations supporting the flexibility-via-subspace hypothesis.

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Acknowledgements

The authors thank members of the Churchland laboratory for reviewing and editing this manuscript. M.M.C. thanks Y. Pavlova for laboratory management. K.V.S. thanks B. Davis for administrative support and S. Kosasih for laboratory management. M.M.C is supported by the Grossman Center for the Statistics of Mind, the Simons Foundation Collaboration on the Global Brain, the Kavli Institute for Brain Science and the Zuckerman Mind Brain Behaviour Institute. K.V.S. was supported by National Institute of Neurological Disorders and Stroke (NINDS) U01NS123101, U19NS112954 and R01NS116623; National Institute on Deafness and Other Communication Disorders (NIDCD) R01NS121097, R01DC014034, U01DC019430 and U01DC017844; National Institutes of Health (NIH) National Institute of Mental Health (NIMH) R01MH086373; the Simons Foundation Collaboration on the Global Brain; L. and P. Garlick; S. and B. Reeves; the Wu Tsai Neurosciences Institute; the Bio-X Institute at Stanford University; The Hong Seh and Vivian W. M. Lim Professorship at Stanford University; and the Howard Hughes Medical Institute (HHMI) at Stanford University.

Author information

Authors and Affiliations

  1. Department of Neuroscience, Columbia University, New York, NY, USA

    Mark M. Churchland

  2. Grossman Center for the Statistics of Mind, Columbia University, New York, NY, USA

    Mark M. Churchland

  3. Kavli Institute for Brain Science, Columbia University, New York, NY, USA

    Mark M. Churchland

  4. Department of Electrical Engineering, Stanford University, Stanford, CA, USA

    Krishna V. Shenoy

  5. Department of Bioengineering, Stanford University, Stanford, CA, USA

    Krishna V. Shenoy

  6. Department of Neurobiology, Stanford University, Stanford, CA, USA

    Krishna V. Shenoy

  7. Department of Neurosurgery, Stanford University, Stanford, CA, USA

    Krishna V. Shenoy

  8. Wu Tsai Neurosciences Institute, Stanford University, Stanford, CA, USA

    Krishna V. Shenoy

  9. Bio-X Institute, Stanford University, Stanford, CA, USA

    Krishna V. Shenoy

  10. Howard Hughes Medical Institute at Stanford University, Stanford, CA, USA

    Krishna V. Shenoy

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  1. Mark M. Churchland
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The authors contributed equally to all aspects of the article.

Corresponding author

Correspondence to Mark M. Churchland.

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Competing interests

M.M.C is an inventor of the MINT decoding approach, which has been licensed to Blackrock Neurotech. K.V.S. served on the Scientific Advisory Boards (SABs) of MIND-X Inc. (acquired by Blackrock Neurotech), Inscopix Inc. (merged with Brucker Nano) and Heal Inc.; served as a consultant/adviser for CTRL-Labs (on the founding SAB; acquired by Facebook Reality Labs in Fall 2019, which is now Meta Platform Reality Labs); and served as a consultant/adviser for and was a co-founder (2016–2023) of Neuralink.

Peer review

Peer review information

Nature Reviews Neuroscience thanks Rune Berg, Matthew Perich and the other, anonymous, reviewer for their contribution to the peer review of this work.

Additional information

Dedication

I dedicate this review to my friend and colleague Krishna Shenoy, who died as we were completing the writing. That act of writing brought our long intellectual relationship full circle. Our final days together were spent much like our initial days: sitting in a room talking about preparatory activity. Pondering the nature of preparatory activity had sent us down the long path of trying to understand the chain of neural events that produces voluntary movement. We saw preparation as the ‘first cog’ in that causal chain. Preparatory activity must somehow have, wrapped up in it, the information necessary to generate time-varying patterns of muscle activity that will soon occur. Trying to figure out how that could work was the first cog in our personal journey — one whose various branches trace back to the initial question that consumed us: what does preparatory activity do? Our adoption and promotion of other ideas — dynamics, null spaces, factors — were a straightforward consequence of our desire to answer that question. We hope that this review brings readers on a similar journey.

Publisher’s note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Glossary

Single-neuron response

The response of a real (or simulated) neuron is notated as \({r}_{n}(t)\), where t indicates time and n indicates we are considering the nth neuron.

Population response

The responses of \(N\) neurons, constituting an \(N\)-dimensional vector \({\boldsymbol{r}}(t)\). Each vector element, \({r}_{n}(t)\), describes the activity of one neuron.

Weight

A number indicating the degree to which one thing (for example, a neuron) influences something else (for example, another neuron). Weights come in many flavours and can summarize many things, including synaptic strengths and the impact of neural activity on muscle activity.

Neural dimension

An N-dimensional vector containing one weight per neuron. Just like the weights themselves, neural dimensions come in multiple flavours.

Weighted sum

Weighted sums of neural activity are central to many analyses that seek to understand network function (real or simulated). If the vector \({\boldsymbol{v}}\) is a neural dimension, one computes \(\mathop{\sum }\limits_{n=1}^{N}{{\boldsymbol{v}}}_{n}{{\boldsymbol{r}}}_{n}(t)\). Using vector notation, the weighted sum \(a(t)={{\boldsymbol{v}}}^{{\rm{T}}}{\boldsymbol{r}}(t)\) is referred to as ‘activity in dimension \({\boldsymbol{v}}\)’.

Subspace

A set of one or more orthogonal neural dimensions. Dimensions are collected into columns of a matrix \({\boldsymbol{V}}\). The vector of weighted sums \({\boldsymbol{a}}(t)={{\boldsymbol{V}}}^{{\rm{T}}}{\boldsymbol{r}}(t)\) is referred to as ‘activity in the subspace of \({\boldsymbol{V}}\) ’.

Activity subspace

A subspace that fully captures neural responses. Suppose we computed a(t) as above (see subspaces). If we can successfully invert this relationship using r(t) ≈ Va(t), then the dimensions in V span the activity subspace.

Dimensionality reduction

A method, such as principal component analysis (PCA), for estimating dimensions that constitute the activity subspace.

Readout

A signal that exits the neural population, such as a descending command for muscle activity (see Fig. 1c). Assuming linearity, the readout is a weighted sum of population responses: \(z(t)={{\boldsymbol{b}}}^{{\rm{T}}}{\boldsymbol{r}}(t)\) (see readout dimensions for how \({\boldsymbol{b}}\) is defined).

Readout dimensions

Each readout has an associated neural dimension, \({\boldsymbol{b}}\), containing weights (one per neuron) used for the readout. Analysis and visualization often employ a lower-dimensional \({{\boldsymbol{b}}}^{{\rm{F}}{\rm{a}}{\rm{c}}}\), with one weight per factor.

Output-potent space

The subspace, spanned by the readout dimensions, where neural activity impacts those readouts. Dimension \({\boldsymbol{v}}\) is ‘output-potent’ if it overlaps with a readout dimension: \({{\boldsymbol{b}}}^{{\rm{T}}}{\boldsymbol{v}}\ne 0\).

Null-space

The subspace, orthogonal to the readout dimensions, where neural activity has no impact on those readouts. Dimension \({\boldsymbol{v}}\) is ‘output-null’ if it is orthogonal to each readout dimension: \({{\boldsymbol{b}}}^{{\rm{T}}}{\boldsymbol{v}}=0\).

Factors

For many recurrent networks, computation can be summarized by considering K factors rather than N neurons (see Fig. 1c), with K \(\ll \) N. Each factor is a weighted sum of activity: xi(t) = \({{\boldsymbol{w}}}_{i}^{{\rm{T}}}\)r(t), where \({{\boldsymbol{w}}}_{i}\) is a neural dimension. Considering all factors: x(t) = WTr(t). \({\boldsymbol{W}}\) reflects network connectivity (see recurrent connectivity).

Neural state

The value of the factors, \({\boldsymbol{x}}(t)\), at a particular moment t. ‘Neural state’ may also refer to the population response, \({\boldsymbol{r}}(t)\). Assuming factors are a valid summary of the population response, these definitions are equivalent.

State space

A method for visualizing the neural state as a point in a space where each axis corresponds to a factor (or some other weighted sum of neural activity).

Neural trajectory

The value of the factors, \({\boldsymbol{x}}(t)\), across a span of time. When plotted in state space, trajectories form traces that describe how activity changes with time (see Fig. 1c, inset).

Manifold

The shape formed by the collection of neural trajectories, defining the possible locations of the neural state within the activity subspace. For example, when cycling at different speeds, the manifold has a tube-like shape (see Fig. 2d).

Tuning

A description of how a neuron’s response reflects a specific quantity, such as a stimulus. In recurrent networks, factors can be the relevant quantities.

Factor tuning

The weight un,i quantifies how the ith factor impacts the nth neuron. The neural dimension \({{\boldsymbol{u}}}_{i}\) quantifies the impact on all neurons. Given a matrix \({\boldsymbol{U}}\) of factor-tuning dimensions, population activity is \({\boldsymbol{r}}(t)\approx {\boldsymbol{U}}{\boldsymbol{x}}(t)\). The approximation reflects non-linearity and spiking variability. \({\boldsymbol{U}}\) reflects network connectivity (see recurrent connectivity), and defines the activity subspace.

Output-potent factors

Factor \({x}_{i}(t)\) is output-potent (that is, impacts the readout) if its factor-tuning dimension, \({{\boldsymbol{u}}}_{i}\), is output-potent (overlaps with one or more readout dimensions).

Output-null factors

Factor \({x}_{i}(t)\) is output-null (that is, does not impact the readout) if its factor-tuning dimension, \({{\boldsymbol{u}}}_{i}\), is output-null (lies within the null-space).

Recurrent connectivity

Synaptic connections that cause activity to flow in ‘loops’. The causal flow is simplified by considering factors (Fig. 1c) and matrices \({\boldsymbol{W}}\) and \({\boldsymbol{U}}\) (see factors and factor tuning). Factors are weighted sums of neural responses: \({\boldsymbol{x}}(t)={{\boldsymbol{W}}}^{{\rm{T}}}{\boldsymbol{r}}(t)\). Responses reflect the factors, \({\boldsymbol{r}}(t)\approx {\boldsymbol{U}}{\boldsymbol{x}}(t)\), completing the loop. In some networks, \({\boldsymbol{W}}\) and \({\boldsymbol{U}}\) can be computed directly from synaptic connectivity.

Factor-level dynamics

In recurrent networks, activity at the present moment drives activity at the next moment. Such dynamics are often best described at the factor level: \(\dot{{\boldsymbol{x}}}\) = f(x) + y. The function f defines a flow-field in state space (see Fig. 1c, inset). y captures inputs from outside the observed system.

Rotational dynamics

Dynamics linking two factors, \({x}_{1}\) and \({x}_{2}\), such that neural states flow from one to the other while preserving their order (see Fig. 2a). Suppose that, for three conditions during preparation, factor \({x}_{{\rm{prep}}}\) takes values \(\{-1,1,2\}\) while factor \({x}_{{\rm{exec}}}\) takes values \(\{0,0,0\}\). After a 90° rotation, activity has left the preparatory subspace, \({x}_{{\rm{prep}}}=\{0,0,0\}\), and entered the execution subspace, \({x}_{{\rm{exec}}}=\{-1,1,2\}\). Depending on the network, rotations might continue or might end after ~90°.

Factor estimation

To estimate factors from measurements of neural activity alone, without knowledge of connectivity, experimenters use dimensionality reduction to estimate the activity subspace and invert the relationship r(t) ≈ Ux(t). For example, one may use x(t) ≈ UTr(t), assuming the estimated U is orthonormal.

Motor neurons

Neurons in the spinal cord that connect directly to a muscle. Each motor neuron spike produces one spike in the muscle fibres it innervates. Motor neurons receive direct (monosynaptic) and indirect (polysynaptic) inputs from motor-cortex neurons.

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Churchland, M.M., Shenoy, K.V. Preparatory activity and the expansive null-space. Nat. Rev. Neurosci. 25, 213–236 (2024). https://doi.org/10.1038/s41583-024-00796-z

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