Comparison of brain–computer interface decoding algorithms in open-loop and closed-loop control

Abstract

Neuroprosthetic devices such as a computer cursor can be controlled by the activity of cortical neurons when an appropriate algorithm is used to decode motor intention. Algorithms which have been proposed for this purpose range from the simple population vector algorithm (PVA) and optimal linear estimator (OLE) to various versions of Bayesian decoders. Although Bayesian decoders typically provide the most accurate off-line reconstructions, it is not known which model assumptions in these algorithms are critical for improving decoding performance. Furthermore, it is not necessarily true that improvements (or deficits) in off-line reconstruction will translate into improvements (or deficits) in on-line control, as the subject might compensate for the specifics of the decoder in use at the time. Here we show that by comparing the performance of nine decoders, assumptions about uniformly distributed preferred directions and the way the cursor trajectories are smoothed have the most impact on decoder performance in off-line reconstruction, while assumptions about tuning curve linearity and spike count variance play relatively minor roles. In on-line control, subjects compensate for directional biases caused by non-uniformly distributed preferred directions, leaving cursor smoothing differences as the largest single algorithmic difference driving decoder performance.

Appendices

Appendix A1: Fisher tuning function

The Fisher tuning function is defined as

$$ f(\theta) = b + c\exp(\kappa\cos\theta), \label{eq:Fisher-tuning} $$

(9)

where θ is the angle between the cell’s preferred direction and the movement direction (Amirikian and Georgopulos 2000). Note that this corresponds to the log-linear model Eq. (6) when b is omitted. The parameters b, c and κ determine the tuning properties. These parameters are translated into three physiological parameters: the half-width of the dropoff shape θ _hw , the modulation depth m _d , and the baseline firing rate b _s , as

$$ \theta_{hw} = \cos^{-1}\Bigg[ \frac{\log(\cosh\kappa)}{\kappa} \Bigg] ~, $$

(10)

$$ m_d = 2c\sinh\kappa, $$

(11)

and

$$ b_s = b + ce^{-\kappa}. $$

(12)

The half width takes θ _hw  < π/2 for κ > 0. For κ→0 (as θ _hw →π/2), Eq. (9) becomes the cosine tuning function,

$$ f(\theta) = b_s + \frac{m_d}{2} + \frac{m_d}{2}\cos\theta. $$

(13)

Appendix A2: Exponential family regression

Here, we briefly describe the exponential family regression and its extension to the state-space method which provides a unifying framework for the neural decoders introduced in Section 2.3. We assume that the spike counts of the ith cell (i = 1,...,N) follows the exponential family distribution,

$$ p(y_{i,t}|\theta_i,\phi_i) = \exp[(y_{i,t}\theta_i - b(\theta_i))/a(\phi_i)+c(y_{i,t},\phi_i)], \label{eq:exp-family} $$

(14)

where θ _i is the canonical parameter related to the mean of y _i,t and ϕ _i is the dispersion parameter (McCullagh and Nelder 1989). This family includes commonly used distributions such as the Poisson and Gaussian. The mean and variance of y _i,t are given as E(y _i,t) = b′(θ _i ) and Var(y _i,t) = a(ϕ _i )b′′(θ _i ), respectively. Since the tuning function f(v _t , p _i ) relates the velocity to the firing rate as E(y _i,t) = f(v _t , p _i ), θ _i  = θ _i (v _t ) is a function of v _t . Assuming that the spiking of N cells are independent of each other, the probability distribution of y _t = (y _{1, t} ,...,y _{N, t} ) is given by

$$ p(\boldsymbol{y}_t|\boldsymbol{v}_t) = \prod\limits_{i=1}^Np(y_{i,t}|\theta_i(\boldsymbol{v}_t),\phi_i). \label{eq:population-distribution} $$

(15)

Let l(v _t ) = log p(y _t |v _t ) be the log likelihood function of Eq. (15). The gradient of l(v _t ) is then derived as

$$ \nabla_{v_t}l(\boldsymbol{v}_t) = \sum\limits_{i=1}^N\frac{y_{i,t}-f(\boldsymbol{v}_t,\boldsymbol{p}_i)}{Var(y_{i,t})}\nabla_{v_t}f(\boldsymbol{v}_t,\boldsymbol{p}_i), \label{eq:loglikeli} $$

(16)

where $\nabla_{v_t}$ denotes the gradient with respect to v _t . The maximum likelihood estimate of v _t is then obtained by solving $\nabla_{v_t}l(\boldsymbol{v}_t) = 0$. Note that only the tuning function f(v _t , p _i ) and the spike count variance Var(y _i,t) appear in Eq. (16). When the tuning function is linear and the variance is constant, $\nabla_{v_t}l(\boldsymbol{v}_t) = 0$ is solved analytically leading to the LGB (i.e. the optimal linear estimator). Otherwise it is solved numerically (by the Newton-Raphson method, for example). Note also that it is seen from Eq. (16) that the maximum likelihood estimate corresponds to the weighted (nonlinear) least-squares estimate which minimizes the weighed squared error,

$$ \sum\limits_{i=1}^N\frac{[y_{i,t}-f(\boldsymbol{v}_t,\boldsymbol{p}_i)]^2}{Var(y_{i,t})}. \label{eq:WLS} $$

(17)

In state-space filtering, in addition to the likelihood of velocity Eq. (16), which is called the observation model, we take a state model describing the evolution of the state, v _t . In our analysis, the state model was taken to be a random walk,

$$ p(\boldsymbol{v}_{t+1}|\boldsymbol{v}_t) \!=\! \frac{1}{(2\pi\eta)^{d/2}} \exp\bigg[\!-\frac{1}{2\eta}(\boldsymbol{v}_{t+1}\!-\boldsymbol{v}_t)^T(\boldsymbol{v}_{t+1}\!-\boldsymbol{v}_t)\bigg]. $$

(18)

The set of posterior distributions of v _t given the observation up to time t, for t = 1,2,..., is then computed by the recursive relationships,

$$ p(\boldsymbol{v}_t|\boldsymbol{y}_1,\ldots,\boldsymbol{y}_t) \propto p(\boldsymbol{y}_t|\boldsymbol{v}_t)p(\boldsymbol{v}_t|\boldsymbol{y}_1,\ldots,\boldsymbol{y}_{t-1}), \label{eq:posterior} $$

(19)

where

$$ p(\boldsymbol{v}_t|\boldsymbol{y}_1,\ldots,\!\boldsymbol{y}_{t-1}) \!=\!\! \int\!\! p(\boldsymbol{v}_t|\boldsymbol{v}_{t-1})p(\boldsymbol{v}_{t-1}|\boldsymbol{y}_1,\ldots,\boldsymbol{y}_{t-1})d\boldsymbol{v}_{t-1} \label{eq:predictive} $$

(20)

is called the predictive distribution. Note that in the limit of large variance, η → ∞, (or when there is no dependence of v _t on v _t − 1) the state-space estimate (more precisely, the maximum a posteriori (MAP) estimate) converges to the maximum likelihood estimate obtained by solving $\nabla_{v_t}l(\boldsymbol{v}_t) = 0$.

One implementation of the recursive formula approximates the posterior distribution Eq. (19) as a Gaussian centered on its mode (Brown et al. 1998). Let v _t|t and V _t|t be the (approximate) mode and covariance matrix for the posterior distribution Eq. (19), and v _{t|t − 1} and V _{t|t − 1} be the mode and covariance matrix for the predictive distribution Eq. (20) at time t. Further, let r(v _t )= log{p(y _t |v _t ) p(v _t |y ₁,...,y _t − 1)}. The posterior distribution is then approximated as a Gaussian whose mean and covariance are $\boldsymbol{v}_{t|t} = \arg\max_{\boldsymbol{v}_t}r(\boldsymbol{v}_t)$ and $V_{t|t} = -[\nabla\nabla_{v_t} r(\boldsymbol{v}_{t|t})]^{-1}$, respectively (where $\nabla\nabla_{\boldsymbol{v}_t}$ represents the second derivative with respect to v _t ). Since the state model is a random walk, the predictive distribution Eq. (20) is also Gaussian, whose mean and covariance are computed as

$$\boldsymbol{v}_{t|t-1} = \boldsymbol{v}_{t-1|t-1}, \label{eq:predictive-mean} $$

(21)

$$V_{t|t-1} = V_{t-1|t-1} + \eta I. \label{eq:predictive-var} $$

(22)

We take the initial state for filtering to be the center of the workspace. This approximate filter is a version of Laplace-Gaussian filter in which the posterior distribution is approximated to be a Gaussian by using Laplace’s method (Koyama et al., unpublished manuscript). Note that we used the Gaussian approximation instead of particle filtering (Doucet et al. 2001) to compute the posterior estimates because under the constraint of computational cost for real-time decoding, the Gaussian approximation provides faster and more accurate posterior estimates than the particle filter (Koyama et al., unpublished manuscript).