# Research

The human brain comprises approximately 100 billion neurons, each with approximately 1,000 connections to other neurons. How do these neurons coordinate to control movement? To discover fundamental principles we conduct electrophysiological experiments with nonhuman primates (NHPs) and people, and create theories and computational analyses to decipher how computations are performed through their collective dynamics (Computation Through Dynamics, CTD; Vyas et al. (2020) Ann Rev Neurosci). Starting in 2006 we proposed a 'Dynamical Systems Framework' (DSF; Shenoy et al. (2013) Ann Rev Neurosci; Shenoy & Kao (2021) Nat Comm) [1], inspired by advances in engineering and by Laurent, Fetz and colleagues, to place studies of cortical motor control on a more mathematical footing. Success depends on the ability to relate single-trial behaviors to single-trial neural activity, estimated on a millisecond timescale. The instantaneous status of neural activity (action potential firing rate) across neurons is termed the 'neural population state,' and the ms-by-ms evolution of this state, which is governed by dynamics arising from synapses and circuits, is termed the single-trial 'neural population trajectory'. The CTD-DSF has shifted how hypotheses are generated and tested, and many groups studying various neural systems have adopted and advanced the CTD-DSF. It has revealed several mechanistic 'dynamical motifs' which appear to be conserved across many areas, tasks and species (mice, NHPs, humans). Our overarching hypothesis is that dynamical motifs are an essential computational building block. The CTD-DSF has helped move systems motor neuroscience from somewhat qualitative and representational interpretations of individual neurons to a quantitative and computationally-mechanistic formulation of neural populations, and does so in a way that bridges to cellular and molecular mechanisms.

We have pursued a deeper understanding of the cortical control of arm movements, motor learning, decision-making and BCI control [2]. We also paired the CTD-DSF with human electrophysiology focused on understanding rapid, dexterous, multi-limb/finger movements and speech, which is only possible with people. We have uncovered new encoding principles (Willett*, Deo* et al (2020) Cell) and these discoveries enable new neuroscience studies and classes of BCIs: a "Brain-to-Text" attempted-handwriting BCI (Willett et al. (2021) Nature) and a "Brain-to-Speech" attempted-speaking BCI (Stavisky et al. (2019) eLife). This 'only possible in humans' neuroscience helps highlight and prioritize new opportunities for next-generation basic science, in NHPs and in people. This human research is a collaboration with Henderson (at Stanford) & Hochberg (at MGH/Brown and across sites) who direct the BrainGate2 pilot clinical trial (NCT00912041).

Despite the importance of understanding the principles and mechanisms underlying motor control, many fundamental questions remain elusive. Due to their centrality in dexterous-movement control, we have focused on dorsal premotor (PMd) and primary motor (M1d) cortex in the arm, hand and finger regions of NHPs and people.

PMd and M1d were among the first cortical areas studied, yet many basic response properties remain poorly understood. It remains controversial whether individual-neuron activity relates to muscles or to abstract movement features. Central to this debate is the complex, multi-phasic and heterogeneous individual-neuron responses. One explanation is that responses represent many movement parameters, though numerous studies have shown that this is merely an approximation. We introduced and are advancing an alternate hypothesis, where motor cortex constitutes a dynamical system that in part supports the dynamics themselves which are needed to control movement. This shifts the field from describing individual-neuron responses in somewhat qualitative terms to quantitatively / mechanistically modeling neural-population activity. This is not single-neuron nihilistic: it does not ignore or attempt to average away the complex features of individual-neuron responses. Rather, by capturing the underlying dynamics it is possible to explain the seemingly idiosyncratic responses.

In its simplest, deterministic form 'neural-population state' is governed by a dynamics equation, **x**(t+1) = f(x(t)) + **B u**(t), where **x**(t) is the neural-population state and is a vector describing the firing rate of all neurons at time t. The next time step is **x**(t+1), f is a nonlinear function, u(t) is an input vector from other brain areas and **B** is a matrix projecting **u**(t) into **x**(t+1). The evolution of **x**(t) reflects circuit dynamics whose purpose is to produce neural signals for preparing and generating accurate movements. Individual-neuron 'tuning' arises incidentally, and is not elemental. The CTD-DSF predicts that dynamical features and motifs should exist in neural-population responses, which so far has held up across areas, tasks and species.

These nonlinear dynamical systems (NLDS) can often be estimated, but it is typically informative to first assess how a simpler linear dynamical system (LDS) performs. We start with a trained behavior, kinematics and action potentials from hundreds of neurons. After identifying each neuron’s location, putative cell-type and directional-tuning preference, DSF parameter estimation begins.

An LDS is described by a dynamics equation and an observation equation. The dynamics equation is **x**(t+1) = **A x**(t) + **B** u(t), where f is replaced by the linear dynamics matrix **A**. The observation equation, **y**(t) = **C x**(t) + **d**, maps x(t) into the firing rate for each measured neuron (**y**(t)) via the matrix **C** and an offset vector **d**. LDS dynamics can be contractive, expansive, rotational or a fixed point causing a variety of possible neural trajectories. Inputs may cause the dynamics to exhibit more complex structure. It is often necessary to engage more complex NLDSs for certain tasks (e.g., decision making, full behaviors that transition between locally-linear regimes). Recurrent neural networks (RNNs) are a powerful approach for implementing f. We have worked to advance RNNs and their adoption, including our new "LFADS" technique. Accurately estimating single-trial trajectories is central to understanding the relationship between ms-timescale neural and behavioral events. These trajectories are highly similar for a given reach direction regardless of the session, providing evidence that the neural circuit dynamics are stable over days/weeks.

[1] Yu et al. (2006) **NeurIPS**; Churchland et al. (2006) **J Neurosci**; Churchland et al. (2010) **Neuron**; Churchland*, Yu* et al. (2010) **Nat Neurosci**; Churchland*, Cunningham* et al. (2012) **Nature**; Mante*, Sussillo* et al. (2013) **Nature**; Kaufman et al. (2014) **Nat Neurosci**; Sussillo et al. (2015) **Nat Neurosci**; Pandarinath et al. (2018) **Nat Meth**; Vyas et al. (2018) **Neuron**; Vyas et al. (2020) **Neuron**; Peixoto*, Verhein* et al. (2021) **Nature**

[2] Santhanam*, Ryu* et al (2006) **Nature**; Gilja*, Nuyujukian* et al. (2012) **Nat Neurosci**; Gilja*, Pandarinath*, et al. (2015) **Nat Med**; Pandarinath*, Nuyujukian* et al. (2017) **eLife**; Willett*, Deo* et al. (2020) **Cell**