Dissociation between sustained single-neuron spiking β-rhythmicity and transient β-LFP oscillations in primate motor cortex

Some of my thesis work has just been published!

Rule et al. 2017 explores the neurophysiology of beta (β) oscillations in primates, especially how single-neuron activity relates to population activity reflected in local field potentials (a.k.a. "brain waves").

β (~20 Hz) oscillations occur in frontal cortex. We've known about them for about a century, but still don't understand how they work or what they do. β-wave activity is related to "holding steady", so to speak.

β is dysregulated in Parkinson's, in which movements are slowed or stopped. β is also reduced relative to slow-wave activity (θ) in ADHD, a disorder associated with motor restlessness and hyperactivity.

I looked at β oscillations during movement preparation, where β seems to play a role in stabilizing a planned movement. We found that single neurons had very little relationship to the β-LFP brain waves. However! This appears to be for a good reason: the firing frequencies of neurons store information about the upcoming movement, and so are diverse and cannot lock to a single frequency.

Anyone who's played in an orchestra knows that when notes are just slightly out of tune, you get interference patterns called beats. The same thing is happening in the brain, where many neurons firing at slightly different "pitches" cause β-LFP fluctuations, even though the underlying neural activity is constant.

This result provides a new explanation for how β-waves can appear as "transients" during motor steady-state: the fluctuations are cased by "beating", rather than changes in the β activity in the individual neurons. This differs from the prevailing theory for the origin of β transients in more posterior brain regions.

Optogenetic Stimulation Shifts the Excitability of Cerebral Cortex from Type I to Type II: Oscillation Onset and Wave Propagation

A new paper by Stewart Heitmann et al. could help us understand what happens when we stimulate cerebral cortex in primates using optogenetics. Modeling how the brain responds to stimulation is important for learning how to use this new technology to control neural activity.

Optogenetic stimulation elicits gamma (~50 Hz) oscillations, the amplitude of which grows with the intensity of light stimulation. However, traveling waves away from the stimulation site also emerge.

It's difficult to reconcile oscillatory and traveling-wave dynamics in neural field models, but Heitmann et al. arrive at a surprising and testable prediction: the observed effects can be explained by paradoxical recruitment of inhibition at low levels of stimulation, which changes cortex from a wave-propagating medium to an oscillator. (Excitation later overwhelms inhibition, giving rise to the observed gamma oscillations.)


Wilson-Cowan neural fields in WebGL

WebGL offers a simple way to write portable general-purpose GPU (GPGPU) code with visualization. The Github repository WebGPGPU walks through a series of examples to bring up a port of Wilson-Cowan neural field equations in WebGL.

The Wilson-Cowan equations are a mean-field approximation of neural activity under the assumption that spiking is asynchronous, and that observations are averaged over a large number of uncorrelated neurons. Wilson-Cowan neural fields have been used to describe visual hallucinations, flicker phosphines, and many other wave phenomena in the brain.

The Wilson-Cowan simulation demo is rudimentary at the moment, but contains a couple presets and rich and diverse dynamics are possibly by varying parameters. For entertainment, there is also an example of a full-screen pattern forming system mapped to (approximate) retinal coordinates using the log-polar transform.

Other cool WebGL implementations pattern-forming systems include the Gray-Scott reaction diffusion equations by pnmeila and Felix Woitzel's reaction diffusion and fluid dynamics simulations. Robin Houston's reaction diffusion implementation is also a good reference example for learning WebGL coding, and Robter Muth's smoothlife implementation is mesmerizing. The hope is that the examples in the WebGPGPU project provide a walk-through for how to build similar simulations in WebGL, something that is not immediately obvious if following existing WebGL tutorials.

Technical notes

I've avoided using extensions like floating point textures that aren't universally supported. Hopefully the examples in WebGPGPU will run on most systems, but compatibility issues likely remain.

Most of the examples store floating point values in 8-bit integers. This works for the firing rate version of the Wilson-Cowan equations because the firing rate is bounded between 0 and 1, so fixed precision is adequate. There are some caveats here. For example, how floating point operations are implemented, and how floats are rounded when being stored in 8-bit values, are implementation dependent. These simulations a quite sensitive to parameters and rounding errors, so different implementations can lead to different emergent dynamics. A partial workaround is to manually control how floats are quantized into 8-bit values. Floats can also be stored in 16 or 32 bit fixed-point precision, at the cost of performance. My understanding is that the default precision for fast floating point operations on the GPU is fairly low, such that 16 bits can capture all of the available precision for the [0,1] range.

The quantization of state variables can lead to numerical errors if the integration step size is too small relative to the timescales of the system. In short, if states are changing too slowly, their increments becomes smaller than 1/256, and numerical accuracy degrades because state changes are rounded-out. Moving to 16-bit precision lessens this issue, but at a cost of an approximately 2-fold slowdown, and so is not implemented except in one example.

Working with the WebGL library feels quite clumsy, because in essence the API exposes the GPU as a state machine. WebGPGPU includes some libraries that hide a lot of the boilerplate and expose a more functional interface for GPU shaders (kernels). For simulations on a grid, there is additional boilerplate to gain access to single pixes. One must tile the viewport with polygons, provide a default fragment shader, and set the viewport and canvas sizes to match the on-screen size. WebGPGPU hides this difficulty, but also walks through this process in the early examples to make it clear what is really happening.

Passing parameters to shaders involves a fair amount of overhead in Javascript, which becomes prohibitive for shaders with a large number of parameters. However, compiling shaders has relatively little Javascript overhead. In practice it is faster to include scalar parameters as #defines rather than pass them as uniforms. WebGPGPU contains some routines to phrase shader compilation as a sort of 'partial evaluation'.

This is a work in progress.


Numerically stable Viterbi algorithm in Python for hidden markov model state inference

The Python demonstration of the Viterbi algorithm on Wikipedia is numerically unstable for moderate to large problem sizes. This implementation is phrased in terms of log probabilities, and so has better numerical properties. Please reuse, creative commons.

def hmm_viterbi(Y,logP,logA,logB):
    See https://en.wikipedia.org/wiki/Viterbi_algorithm

    Y : 1D array
        Observations (integer states)
    logP : array shape = (nStates ,)
        1D array of priors for initial state
        given in log probability
    logA : array (nStates,nStates)
        State transition matrix given in log probability
    logB : ndarray K x N
        conditional probability matrix
        log probabilty of each observation given each state
    K = len(logP)         # Number of states
    T = len(Y)            # Number of observations
    N = np.shape(logB)[1] # Number of states
    Y = np.int32(Y)

    assert np.shape(logA)==(K,K)
    assert np.shape(logB)==(K,N)

    # The initial guess for the first state is initialized as the
    # probability of observing the first observation given said 
    # state, multiplied by the prior for that state.
    logT1 = np.zeros((K,T),'float') # Store probability of most likely path
    logT1[:,0] = logP + logB[:,Y[0]]

    # Store estimated most likely path
    T2 = np.zeros((K,T),'float')

    # iterate over all observations from left to right
    for i in range(1,T):
        # iterate over states 1..K (or 0..K-1 with zero-indexing)
        for s in range(K):
            # The likelihood of a new state is the likelihood of 
            # transitioning from either of the previous states.
            # We incorporate a multiplication by the prior here
            log_filtered_likelihood = logT1[:,i-1] + logA[:,s] + logB[s,Y[i]]
            best = np.argmax(log_filtered_likelihood)
            logT1[s,i] = log_filtered_likelihood[best]
            # We save which state was the most likely
            T2[s,i] = best

    # At the end, choose the most likely state, then
    # Iterate backwards over the data and fill in the state estimate
    X     = np.zeros((T,) ,'int'  ) # Store our inferred hidden states
    X[-1] = np.argmax(logT1[:,-1])
    for i in range(1,T)[::-1]:
        X[i-1] = T2[X[i],i]
    return X


Better 3D graphics on the Arduino: avoiding flickering and tearing

A while ago I purchased a cheap $4 Chinese LCD Arduino shield from Ebay (e.g 1 2 3). The board arrived with no documentation. Disassembling the shield revealed no ICs, indicating that the driver is integrated with the LCD itself. Upon request, the vendor provided an archive containing a few amusingly translated datasheets, as well as a copy of Adafruit's Arduino LCD drivers. Evidently the product is a clone of the Adafruit LCD shields, and uses the ILI9341 LCD driver. I had hoped to use the display to show short animated GIF loops. This is not, in practice, possible. Test animations loaded slowly, with noticeable flicker and vertical tearing. The Arduino does not have enough speed or bandwidth to render full-screen animation frames, but what about 3D vector graphics?


Wilson-Cowan neural field model, in HTML/JavaScript, in under 512 bytes

This started essentially as a bet made at a bar. It turns out that one can make a Wilson-Cowan neural field demo in under 512 bytes in HTML/JavaScript, so long as one is willing to use invalid HTML that will only render in Firefox.

View demo hosted on Dropbox (Firefox only)

For these parameters, the simulation starts out with noise. Radiating waves slowly form. Eventually the solution converges to spiral waves.

Here's the source. There are 466 bytes of actual JavaScript and and 491 bytes total.

In addition to refactoring expressions to reduce size and using invalid HTML, the code also takes advantage of the facts that
  • indecies into a 256 x 256 array can be wrapped horizontally and veritcally by masking with 0xffff, yielding periodic boundary conditions for free
  • a succinct way to get a grayscale canvas on a white page is to just write to the alpha bytes of the canva's image data
  • "for(i=N;i--)" iterates over 0 to N-1 in reverse order, and is shorter than "for(i=0;i<N;i++)"
It's actually very readable if you add whitespace and comments back in.
c=document.body.lastChild// get the canvas (it's the only element)
c.width=c.height=W=256   // set canvas size (and width constant)
Z=c.getContext('2d')     // get graphics context
l=Z.getImageData(0,0,W,W)// get handle to image data buffer
N=W*W                    // define a constant the length of the data buffer
I=()=>Float32Array(N)    // define short function for initializing buffers
t=I()                    // make scratch for convolution intermediate values
b=I()                    // this will be the E-cell buffer
f=I()                    // this will be the I-cell buffer
g=I()                    // buffer to hold blurred E-cell state
h=I()                    // buffer to hold blurred I-cell state
K=(w,t,W)=>{             // function for separable convolution.
                         // blurs from NxN buffer t into 256x256 buffer w
                         // W is the step size. If it is 1, the blur will be
                         // horizontal. If it is 256, the blur will be 
                         // vertical. Gaussian is convolution is separable
                         // in x/y, so we can do the 2D blur by first 
                         // doing the x and then the y
    for(i=N;i--;){       // iterate over all points
        w[i]=0;          // initialize accumulator to zero
        for(j=3;j--;)    // iterate over kernel
                         // The expression (j*4+2&7) saves one character over
                         // the more readable [2,6,2][j]/5
                         // we wrap indecies in x and y by masking with
                         // 0xffff i.e. N-1
    K(t,b,1)             // convolve E-cell buffer in x direction
    K(g,t,W)             // convolve E-cell buffer in y direction
    K(t,f,1)             // convolve I-cell buffer in x direction
    K(h,t,W)             // convolve I-cell buffer in y direction
    for(i=N;i--;){       // for every point
                         // B is an update function
                         // includes firing rate nonlinarity and some of the
                         // Wilson-cowan update -- expression have been
                         // refactored to minimize size
        b[i]=B(6*h[i]-g[i])+b[i]*.3 // update E-cell field
        f[i]=B(1)*.13+f[i]*.91      // update I-cell field
        l.data[i*4+3]=b[i]*(W-1)    // store E-cell value in alpha channel
    Z.putImageData(l,0,0);          // update image
    setTimeout(w,0)                 // start next frame
w()                                 // start simulation