Review 13: Computation Through Neural Population Dynamics Part 1

Computation Through Neural Population Dynamics by Saurabh Vyas

Citation: Vyas, Saurabh, et al. “Computation through neural population dynamics.” Annual Review of Neuroscience 43 (2020): 249-275.

  • Gonna make this a two-part review since this one is a little longer.
  • The dynamic systems approach to population neuron activity is interesting. I just recently learned about the field called dynamical systems, which is exciting. I had always thought “dynamical system” was an adjective (and it is), but I never imagined they have their own field of mathematical research.
    • formal def. of dynamical system: $\frac{dx}{dt} = f(x(t), u(t)) $.
    • $x$ is the neural population response at time $t$ and $u$ is the external input.
    • pendulum as a brief example of a dynamical system.
  • Move from introduction to the brain as a dynamical system to discussing neuronal population firing trajectory.
    • Here they bring up dimensionality reduction as an efficient method for simplifying the case when you have thousands of neuron trajectories and of course this is a valid approach. However, the statement that “we only lose $x$ amount of variance with PCA, so PCA is good” is one I take with a grain of salt. Most data can have most of its variance accounted for by 10 PCs. I don’t think accounted variance means dim. reduction on neural trajectories is a great idea. On the other hand, it’s not likely we’re gonna start looking at all trajectories and find intuitive understanding, so I’ll stop complaining.
  • Computation through Dynamics (CTD) : dynamics of how neural circuit + input evolve through state space.
    • underlying all the things we do!
  • CTD mathematical intro
    • Deep Learning
      • Data modeling: use the internal state to find $f$
      • Task modeling: input / output pairs from a task and we wish to identify $f$ that could have possibly led to these pairs.
        • Task modeling seems so much harder!
      • RNNs as the neural network / DL approach to solving the $f$ responsible for dynamical systems.
        • glad they used the pendulum example to explain this.
    • High Dim. Systems
      • Linear dynamical system (LDS) : $\frac{dx}{dt} = L(x(t),u(t)) = Ax(t) + Bu(t)$
      • Linear dynamics around fixed points: attractor, repeller, oscillator.
      • Stable orbits / unstable orbits.
      • LDS around fixed point with input: $\frac{d\delta (s)}{dt} = A(x^*,u^*)\delta x(t) + B(x^*,u^*)\delta u(t) $
        • $\delta x(t) = x(t) - x^*$ and same for $\delta u(t)$. This just means they have an initial point denoted by $*$ term.
    • Linear Subspaces and Manifolds
      • 2D manifold looks like a pringle.
      • Most of this was reviewed, but I did not understand the purpose of figure 6 showing potent space and null space.
  • Dynamical Systems in Motor Control
    • What principles guide the generation of the observed time-varying patterns of motor cortical activity?
    • Delay reaching task as an example of a common experimental paradigm for motor preparation.
      • CTD views preparatory activity as an initial condition. Not a necessary condition though; Other initial states can have the same motor movement, but being closer to the right one is advantageous for several reasons that they explain.
        • Competing hypothesis presented.
      • RNN model to try to produce motor activity given preparatory activity.
      • Null spaces as preparation preclude some activity because it is orthogonal to that initial condition.
    • Dynamical motifs during movement
      • Condition invariant signal: signal that follows timing/onset of cue. Large component of population response.
        • Roughly, releases the preparatory state from an attractor-like state to a rotating state.
        • The paper with citation (Sussile 2015) that is a “sufficiency test for the rotations motif” seems really interesting. Simple low dim RNN oscillation produced complex inputs/outputs for prep state –> muscle activity.
      • A really interesting point is that maybe these oscillating states are inevitable outcomes of basic neuronal firing properties.
        • I don’t think asking “when are these oscillating states informative?” is as clear as asking “how informative are these oscillation states?”.
      • Two similar states –> same movement == tangling. Then the reverse process is untangling.
      • Low degree of trajectory divergence in supplementary motor area (SMA).
      • Summart lost me a (d) - (f). Feels like I missed those later points somewhere in the reading.