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Nicolelis MAL, editor. Methods for Neural Ensemble Recordings. 2nd edition. Boca Raton (FL): CRC Press; 2008.

## INTRODUCTION

The advance of BMIs was largely motivated by investigations of velocity encoding in single neurons during stereotypical reaching experiments. However, BMIs are designed to decode neural activity from an ensemble of neurons and direct general reaching movements. Hence, neural data analysis strategies for BMIs are required to: (1) analyze the neural activity from ensemble of neurons, (2) account for the dynamical nature of the neural activity associated with general reaching movements, and (3) explore and exploit other relevant modulating signals.

Decoding neural activity can be performed either in a single stage or two stages. Two-stage decoding relies on a preliminary encoding stage to determine how the neurons are tuned to the relevant biological signals. Based on the estimated tuning curves, the neural activity across an ensemble of neurons can be decoded using either a population-vector, maximum likelihood estimation, or Bayesian inference (Pouget et al., 2003). The population-vector approach results in a linear relationship between the spike counts and the estimated biological signal, which can be estimated directly in a single stage using linear regression (Brown et al., 2004). This chapter focuses on single-stage decoding with linear regression, and in particular on two special challenges facing the application of linear regression to neural ensemble decoding during reaching movements (see “Movement Prediction”). First, given the dynamic nature of the decoded signal, it is necessary to include the history of the neural activity, rather than just its current spike count. Second, due to the correlation between the activities of different neurons (see “Ensemble Analysis”) and the activities in different time lags, the resulting regression problem is ill-posed and requires regularization techniques (see “Linear Regression”).

Although neural decoding can be performed in a single stage, neural encoding is still important for investigating which signals are encoded in the neural activity. For this purpose, the notion of tuning curves is generalized to characterize how the neural activity represents the spatiotemporal profile of the movement. This analysis quantifies both the spatiotemporal tuning curves and the percent variance of the neural activity that is accounted by the movement profile (see “Neuronal Encoding and Tuning Curves”). For comparison, the percent variance in the neural activity that might be related to general neural modulations is assessed independently under the Poisson assumption (see “Neuronal Modulations”). These two-faced variance analyses provide a viable tool for quantifying the extent to which the neural code is effectively decoded, and the potential contribution of yet undecoded modulating signals.

The strategies and algorithms described in this chapter are demonstrated using the neural activity recorded from an ensemble of cortical neurons in different brain areas during a typical target-hitting experiment with pole control as described in Carmena et al., 2003.

## NEURONAL ENCODING AND TUNING CURVES

The firing rates of cortical motor neurons represent a diversity of motor, sensory, and cognitive signals, and most notably the direction and speed of movement (Georgopoulos, 2000; Johnson et al., 2001; Georgopoulos et al., 1989; Paz et al., 2004). Neuronal encoding of specific movement-related signals, including movement direction, and speed, has been characterized in terms of the tuning properties of the neurons (Georgopoulos et al., 1986, Ashe and Georgopoulos 1994). Most prominently, center-out reaching experiments indicated that the firing rates of single cortical motor neurons are broadly “tuned” to the direction of movement. Tuning curves represent the firing rate as a function of the direction of movement, and are well described by a cosine function of the angle between the movement direction and the preferred direction of the neuron. Detailed investigations suggest that the activity of directionally tuned cortical motor neurons is also modulated by the speed of movement, both independently from and interactively with the direction of movement (Moran and Schwartz, 1999). Furthermore, other scalar signals, including the amplitude of movement, its accuracy, the location of the target, and the applied force may also contribute to the firing rate modulation of cortical motor neurons (Johnson, et al., 2001, Alexander & Crutcher, 1990, Georgopoulos et al., 1992, Scott 2003).

BMI experiments can be used to further investigate (Nicolelis 2001; 2003): (1) how individual neurons represent the spatiotemporal profile of the movement during free arm movements; (2) the potential modulations by yet undecoded signals; and (3) the distribution of correlated activity across an ensemble of neurons. This section focuses on the first issue whereas the last two are addressed in the sections “Neuronal Modulation” and “Ensemble Analysis,” respectively.

### Velocity Tuning Curves

It is customary to determine the tuning of motor neurons to the velocity of movement during planar center-out reaching movements, where the direction of velocity is approximately constant during each reaching movement (Georgopoulos, 1986; Moran and Schwartz, 1999). Tuning curves that account for both the cosine-directional sensitivity and the effect of the speed of movement are of the form

where *N* is the number of spike counts during the movement, *V** _{m}* and θ are the magnitude (speed) and direction of the velocity,

*a*and θ

*are the magnitude and preferred-direction of the directional tuning,*

_{PD}*b*is the magnitude of the tuning to the speed,

*c*is the mean spike count across different directions, and is the modeling error. This formulation implies that the neural activity depends linearly on the x- and y-components of the velocity,

*V*

*=*

_{x}*V*

*sinθ and*

_{m}*V*

_{y}*V*

*= cosθ, as:*

_{m}where *a** _{x}* =

*a*sinθ

*and*

_{PD}*a*

*=*

_{y}*a*cosθ

*are the tuning to the components of the movement. Equation 4.2 is in the form of a linear regression, so the tuning coefficients*

_{PD}*a*

*and*

_{x}*a*

*can be estimated directly using linear regression between the neural activity and the velocity. The resulting coefficient of determination*

_{y}*R*

^{2}(

*N*,

*V*) describes the fraction of the total variance in the spike counts that is attributed to the velocity and provides a measure of the goodness of fit. Because the neural activity is highly noisy (see the section “Neuronal Modulation”), the resulting coefficients of determination are small, even if the tuning to the velocity is significant.

Free arm movements involve temporal patterns, which cannot be captured only by spatial features. The formulation of the directional tuning should be generalized to describe the spatiotemporal features of the movement profile. The neural activity is binned (typically with 100 ms long bins), and the mean velocity in each bin is computed. The tuning of the neural activity to the velocity at a particular lag is assumed to follow the cosine tuning of Equation 4.2 (Lebedev et al., 2005) as:

where *N*(*k*) is the spike counts in the *k*th time-bin,*V** _{x}* (

*k + l*),

*V*

*(*

_{y}*k + l*), and

*V*

*(*

_{m}*k + l*)are the mean components and speed of the velocity in the (

*k+l)*th time-bin,

*l*is the relative lag between the velocity and the spike counts (positive or negative

*l*corresponds to rate-modulations preceding or succeeding the velocity measurement, respectively),

__A__*= [*

_{l}*a*

*(*

_{x}*l*)

*a*

*(*

_{y}*l*)] is the vector of directional tuning parameters with respect to the lagged velocity,

*b*(

*l*) is the speed tuning parameter,

*c*(

*l*) is a bias parameter, and (

*l*,

*k*) is the residual error. The coefficient of determination of the single-lag regressions

*R*

^{2}(

*N*,

*V*([

*l*])) for

*l*= −

*L*

_{1},…

*L*

_{2}, and the strength of the tuning |

*(*

__A__*l*)| can be used to evaluate the strength of tuning and determine the most significant lag, i.e., the lag between the neural activity and the velocity that it is tuned to the most.

### Spatiotemporal Tuning Curves

To account for the dependence of the neural activity on the velocity profile across several lags, the model is extended according to the multi-lag regression given by (Figure 4.1):

where *L*_{1} and *L*_{2} are the number of preceding and succeeding lags between the spike counts and the velocity, respectively. This model describes the tuning of the neural activity to the complete velocity profile in the surrounding window of [−*L*_{1} *L*_{2}]. The coefficient of determination of the multi-lag regression *R*_{2} ( *N*,*V* ([−*L** _{1}*,

*L*

*])) can be interpreted as the fraction of variance in the binned spike counts that is attributed to modulation by the spatiotemporal velocity profile. Expressed as a percentage, it is referred to as the percent velocity modulation (PVM).*

_{2}It is noted that because the velocity is slowly varying (compared with the bin size), the velocities in different lags are highly correlated. Thus, the multi-lag regression analysis of Equation 4.4 and the lag-by-lag analysis of Equation 4.3 would yield different regression parameters, as demonstrated below. The multi-lag regression analysis accounts for the correlation between the velocities in different lags and describes the tuning to the complete velocity profile. Furthermore, *R*_{2} (*N*, *V* ([−*L** _{1}*,

*L*

*])) cannot be approximated by the sum of the individual coefficients of determination of the single-lag regressions*

_{2}*R*

^{2}(

*N*,

*V*([

*l*])) for

*l*= −

*L*

_{1},…

*L*

_{2}. Thus, it is necessary to perform the multi-lag regression in order to quantify the PVM.

The multi-lag regression of Equation 4.4 can be formulated in a matrix notation

where *N* = [*N*(*L*_{1} + 1)…*N*(*T* − *L*_{2}) ]* ^{T}*, and

__V__*(*

_{k}*l*) = [

*V*

*(*

_{k}*L*

*+ 1 +*

_{1}*l*)...

*V*

*(*

_{k}*T*−

*L*

_{2}*+ l*)]

*(the index*

^{T}*k*=

*x*,

*y*,

*m*indicates the x- and y- components of the velocity and its magnitude, respectively) are (

*T*−

*L*

_{1}−

*L*

_{2}) × 1 vectors of spike counts and velocity components, respectively,

*1*is a (

*T*−

*L*

_{1}−

*L*

_{2}) × 1 vector of 1’s, and

*= [*

__C__*a*

*(−*

_{x}*L*

_{1})

*a*

*(−*

_{x}*L*

_{1}+ 1)…

*a*

*(*

_{x}*L*

_{2})

*a*

*(−*

_{y}*L*

_{1})…

*a*

*(*

_{y}*L*

_{2})

*b*(−

*L*

_{1})…

*b*(

*L*

_{2})

*c*]

*is a vector of regression coefficients.*

^{T}The optimal least-square solution of Equation 4.5 is sensitive to measurement noise and become unstable when the condition number of the matrix is large (see the section “Least Square Solution”). During typical experiments the condition number of the matrix *X** _{V}* is on the order of 10

^{6}(Figure 4.4). Thus proper evaluation of Equation 4.5 requires regularization methods as detailed in the section “Regularization Methods.” The regression parameters of a typical spatiotemporal tuning curve, computed using regularization is shown in Figure 4.2, and compared with the lag-by-lag tuning curve. The parameters

*a*

*and*

_{x}*a*

*are shown separately as a function of the lag (top left and right, respectively), where negative lags are preceding and thus predictive, whereas positive lags are succeeding and thus reflective. The magnitude of the directional tuning, i.e.,*

_{y}and the estimated preferred direction (PD) at each lag, *PD*(*l*) = tan^{−1}(*a** _{y}*/

*a*

*) are depicted in the bottom left and right panels, respectively. The results based on the lag-by-lag tuning analysis defined by Equation 4.3 are shown for comparison (the values of the resulting coefficients are scaled down by a factor of 5 to compensate for the smaller number of coefficients in each regression). It is evident that the lag-by-lag analysis provides only a coarse and highly smoothed estimate of the underlying multi-lag tuning curve. The estimates of the PDs are reliable only in lags where the magnitude of the directional tuning is large and, thus, the fluctuations in other lags are meaningless. For the lags in which the directional tuning is significant, the estimated PD is the same independent of the method and relatively constant across the lags.*

_{x}The spatiotemporal tuning curve expressed by Equation 4.4 describes how the neural activity is modulated by the velocity profile, and the associated regression analysis quantifies the percent variance attributed to the velocity modulation (PVM). The distributions of the PVM across an ensemble of (183) neurons recorded during a typical target-hitting experiment are depicted in Figure 4.3. Only few neurons exhibit velocity modulations in excess of 20% of their variance. On average, the velocity-profile accounts for only 3.7% of the total variance, and for half of the neurons it accounts for less than 1.8%.

### Tuning to Spatiotemporal Patterns of Velocity

The regularization methods used to compute the spatiotemporal tuning curve are based on decomposing the spatiotemporal velocity profile, given by the matrix *X** _{V}*, into its principal components (PCs) (see the section “Principal Component Analysis (PCA)”). The PCs of the velocity are uncorrelated linear combinations of the lagged velocity components. The first PC accounts for maximum fraction of the variance in the velocity profile, and the succeeding PCs account, in order, for the maximum fraction of the remaining variance. Figure 4.4 shows the variance accounted for by individual PCs, and the accumulated variance accounted by all the initial PCs. It is evident that the late PCs account for a negligible fraction of the total variance. In particular, the first 25 PCs in this example already account for 95% of the variance.

The weights of the linear combinations that define each of the PCs, are the principal directions of the covariance matrix of the velocity profile, and represent the principal velocity-patterns (see the section “Principal Component Analysis (PCA)”). The initial principal velocity-patterns in a 1.9 s window during a typical target-hitting experiment are shown in Figure 4.5. Some of the principal velocity-patterns are easily interpreted in terms of directional velocity and acceleration: The first two principal patterns describe low-frequency directional acceleration and directional velocity, respectively. The third principal pattern describes a higher-frequency directional acceleration, with zero center velocity preceded and succeeded by negative and positive directional velocity, respectively. The fourth principal pattern describes a pause between movements in the same direction. The fifth principal pattern describes low-frequency speed, whereas the sixth describes a change in direction of movement. Other components describe higher frequencies of the velocity and acceleration profiles, including high-frequency speed (8th and 16th components) and higher-frequency acceleration (12th component).

The correlated velocity components that compose the matrix *X** _{V}* (Equation 4.5) can be described alternatively by the uncorrelated velocity PCs. Each velocity PC describes the temporal evolution of the corresponding principal velocity-pattern defined. The correlations of the spike counts with these PCs represent tuning to spatiotemporal patterns of velocity. For example, the tuning of an M1 neuron to the principal velocity-patterns depicted in Figure 4.5, is shown in Figure 4.6. This neuron seems to be tuned to low-frequency directional acceleration (1st component) and velocity (2nd component), and high-frequency speed (12th component). However, the neuron is not selectively tuned to any single spatiotemporal velocity pattern.

## NEURONAL MODULATIONS

The velocity profile accounts for a fraction of the total variance in the spike counts of cortical neurons during arm movements, as suggested for example in Figure 4.3. The remainder of the variance may be attributed to either (1) the neuronal noise associated with the underlying firing activity or (2) other modulating signals, including nonlinear velocity effects. These two possible affects can be differentiated by quantifying the percent variance that is attributed to the neuronal noise, while the remaining variance, is attributed to the other modulating signals.

Statistically, neural spike trains are customary analyzed using the mathematics of point processes (Perkel and Bullock, 1968; Dayan and Abbott, 2001), where each point represents the timing of a spike. The simplest point process, the Poisson process is memory-less, i.e., the probability of spike occurrence is independent of the history of the spike train. The simplest Poisson process, the homogenous Poisson process, is characterized by a constant instantaneous spike rate, and thus is inadequate for describing firing rate modulations. Thus, the simplest point process that can describe rate modulations is the inhomogeneous Poisson process, which is characterized by time-varying instantaneous spike rate that is independent of the history of the spike train (Dayan and Abbott, 2001, Snyder 1975; Johnson 1996). Inhomogeneous Poisson processes in which the instantaneous spike rate is itself a stochastic process are referred to as *doubly stochastic Poisson processes* (Snyder 1975; Johnson 1996). In doubly stochastic Poisson processes, the probability of spike occurrence given the instantaneous rate is described by Poisson statistics. The spike trains recorded during arm movements can be considered as realizations of doubly stochastic Poisson processes because the instantaneous rate depends on a number of biologically relevant stochastic signals. These signals include, for example, the position and velocity of the arm and the muscle forces. These multiple signals affect the instantaneous rate according to the individual tuning of each neuron.

Assuming that the spike trains are generated by doubly stochastic Poisson processes facilitates the analysis of their statistics, which is determined by two factors: stochastic changes in the instantaneous firing rate and Poisson probability of spike occurrence. The distribution of spike counts, *N** _{b}* in bins of size

*b*, is determined by the average instantaneous spike rate during the bin, Λ

*, and its statistics are related to the statistics of Λ*

_{b}*according to (Snyder 1975; Appendix):*

_{b}The last relationship can be interpreted as a decomposition of the total variance in the spike counts into the variance of the underlying information bearing parameter, or rate-modulating signal, *Var* [Λ* _{b}*], and the variance that would occur if

*N*

*was generated by a homogenous Poisson process,*

_{b}*E*[

*N*

*]. Thus, the variance of the rate-modulating signal is the excess variance of the spike counts beyond that of a homogeneous Poisson process. Furthermore, as the instantaneous rate of a homogeneous Poisson process does not vary in time, it cannot be modulated by any relevant biological signal and consequently its variance may be considered as noise. Thus, the signal-to-noise ratio (SNR) can be defined as:*

_{b}where F is the Fano factor, defined as the ratio between the variance and the mean of the spike count (Dayan and Abbott, 2001). In addition, the percent overall modulation (*POM*) is defined by expressing the variance of the rate-modulating signal as a percentage of the variance in the spike counts:

Compared to the Fano factor, the POM emphasizes the variability due to the underlying stochastic modulations and distinguishes it from the high variability inherent in the Poisson process (Zacksenhouse et al., 2007a).

The POM is zero for the homogeneous Poisson process and positive for the inhomogeneous Poisson process. However, if the underlying point process is not Poisson, the variance of the spike counts may be smaller than the mean, and the POM may be negative. It should be noted that negative values may also result from finite-length spike trains due to the variance of the estimate, as demonstrated in the following text.

The distribution of the POM in the ensemble of 183 neurons analyzed in Figure 4.3 is depicted in Figure 4.7 (top). The POM of the recorded neurons was positive for 83% of the neurons in this ensemble. The distribution of the POM for simulated homogeneous Poisson processes having the same mean spike counts as the recorded neurons is shown in the bottom left panel, indicating that the standard deviation of the estimated POM is σ* _{POM}* = 1.2%. The POM of the recorded neurons (Figure 4.7, top) was above 2σ

*= 2.4% for 65% of the neurons, whereas only 6.5% of the neurons exhibited negative POM below −2σ*

_{POM}*= −2.4%.*

_{POM}The POM provides a scale against which the PVM can be compared in order to determine the relative role of the velocity profile in modulating the firing rate. The scatter plot in Figure 4.8 depicts the correlation between the PVM and POM for the same ensemble of (183) neurons during the experiment analyzed in Figure 4.3 and Figure 4.7. The high correlation indicates that the activity of cortical neurons, which exhibited larger rate modulations, was, in general, better correlated with the velocity profile. As expected, the PVM is usually smaller than the POM, in agreement with the interpretation that the POM describes the percent variance attributed to overall modulations, including the velocity modulation. The slope of the linear relationship is only 0.22, suggesting that the PVM accounts for only a small fraction of the POM, and that additional signals, other than the velocity profile, modulate the neural activity (Zacksenhouse et al., 2007).

The POM of the recorded neurons can be also compared to the POM of simulated neurons that are modulated only by the velocity profile (bottom right). The simulated neural activity is generated using inhomogeneous Poisson processes with a rate parameter derived from Equation 4.4 based on the estimated multi-lag tuning curves of the recorded neurons. When the activity is modulated only by the velocity profile, the statistics of the resulting POM distribution is similar to that of the PVM (Figure 4.3), but smaller than that for the recorded neurons. This comparison supports the above conclusion that the POM captures additional signals that modulate the neural activity.

## MOVEMENT PREDICTION

Neuronal analysis indicates that the spike counts generated by individual neurons encode the planned velocity, as detailed in the section “Neuronal Encoding and Tuning Curves.” BMI technology is based on decoding the neuronal activity and extracting the movement related signals, and in particular the planned velocity. Several decoding techniques are possible, including, linear (Weiner) filter, Kalman filter, and nonlinear filters. Typical BMI experiments indicate that the Kalman and nonlinear filters do not consistently outperform the linear filter (Carmena et al., 2003; Wessberg et al., 2000). Here we concentrate on describing the linear filter and its improvement using regularization methods, and demonstrated its superior performance.

The movement signal of interest, like the velocity components, can be predicted by a linear filter of the spike counts recorded from an ensemble of neurons during the preceding time lags, according to:

where (*k*) is the predicted movement signal (e.g., the components of the velocity *V** _{x}* and

*V*

*, or the grip force), ω*

_{y}_{0}is the bias term, and ω

*(*

_{j}*l*) is the weight given to the spike counts elicited by the

*j*-th neuron during the preceding

*l*-th lag. The filter described in Equation 4.9 is of the form of a moving average across multiple neurons, with a window determined by the number of lags

*L*.

The bias and weights are determined from the training section of the BMI experiment, in which both the neural activity and the movement signals are recorded, using the multi-variables regression given by:

where *M*(*k*) is the recorded movement signal, and (*k*) is the residual error.

The multi-lag multineuron regression of Equation 4.10 can be formulated in a matrix notation as:

where, * M* = [

*M*(

*L*+ 1)…

*M*(

*T*)]

*and*

^{T}

__N__*(l) = [*

_{j}*N*

*(*

_{j}*l*)…

*N*

*(*

_{j}*T*− + l −

*L*)]

*are (*

^{T}*T*−

*L*) × 1 vectors of the measured movement-signal and the properly lagged spike counts of the

*j*th neuron, respectively, 1 is a (

*T*−

*L*) × 1 vector of 1’s,

*= [ω*

__W___{1}(

*L*) ω

_{1}(

*L*− 1)… ω

_{1}(1) ω

_{2}(

*L*)… ω

_{2}(1) ω

*(*

_{n}*L*)… ω

*(1) ω*

_{n}_{0}]

*is a vector of regression coefficients, and*

^{T}*n*is the number of neurons.

The coefficient of determination of the multi-variable regression of Equation 4.11, *R*_{2} ( *M*, *N* ([−*L*,1])), describes the fraction of variance in the measured movement-signal that is correlated with the linear filter and provides a measure for the fidelity of the reconstruction. However, in movement prediction we are interested in the ability to predict the movement-signal from new measurements of neural activity. Thus, the performance of the filter is assessed using testing records, which were not used to determine the filter coefficients. The quality of the prediction is evaluated on a testing record using the filter coefficients determined from a nonoverlapping training record. The fidelity of the prediction can be assessed using either (1) the coefficient of regression *R*( *M*, ) between the predicted movement-signal and the measured signal during the testing record *M*, or (2) The variance reduction *VR*( *M*, ) given by

The optimal least-square solution of Equation 4.11 is sensitive to measurement noise and may become unstable when the condition number of the matrix is large (see the section “Least Square Solution”). During typical BMI experiments, the condition number of the matrix *X** _{N,L}* is on the order of 2000. The effect of different regularization parameters on the fidelity of velocity prediction, based on a 10 min training record and a 2 min testing record taken from a typical BMI experiment, is shown in Figure 4.9. It is evident that prediction can be improved by using proper regularization. The best coefficient of determination is obtained with λ = 94, resulting in

*R*

^{2}

_{λ = 94}(

*M*, ) = 0.676 (i.e.,

*R*

_{λ = 94}(

*M*, ) = 0.82), implying that the predicted velocity (depicted in Figure 4.10, middle panel) accounts (explains) 67% of the variance of the measured velocity (Figure 4.10, top panel). In comparison, the least square (LS) regression, corresponding to λ = 0, results in

*R*

^{2}

*(*

_{LS}*M*, ) = 0.593 (i.e.,

*R*

*(*

_{LS}*M*, ) = 0 77). Thus, with proper regularization, the prediction can capture an additional 8% of the variations in the velocity.

The performance of a Kalman filter (Brown and Hwang, 1997; Wu et al., 2004) that was trained and tested on the same records of data is shown for comparison in the bottom panel of Figure 4.10. The resulting coefficient of correlation of *R*^{2} ( *M*, ) = 0.6 (i.e., *R*( *M*, ) = 0 77) indicates that the Kalman filter performs as well as the least-squares linear regression but underperforms the regularized linear regression.

## ENSEMBLE ANALYSIS

### Principal Neurons

The activity of the recorded neurons may be correlated either due to common modulating signals or due to correlation in the neural noise. Ensembles of neurons with correlated activity can be identified using principal components analysis (PCA), as detailed in the section “Principal Component Analysis (PCA).” PCA transforms the sequences of normalized spike counts recorded from *n* neurons into an ordered set of *n* uncorrelated sequences, known as principal components (PCs). Each PC is the weighed linear combination of the *n* normalized (zero mean and unit variance) spike count sequences, and thus may be considered as the normalized activity of a principal neuron. The PCs are ordered according to their variance, with the first PC accounting for most of the variance. The associated unit-length weight vector is the first eigen vector of the covariance matrix of the neural activity, and describes the ensemble of neurons whose superposition carries most of the variance. Each subsequent PC accounts for the maximum of the remaining variance in the neural activity.

The percent variance carried by the different PCs, or principal neurons, during a typical session of a target-hitting experiment is described in Figure 4.11. The percent variance drops significantly for the first few principal components before reaching an approximately constant, nonzero, level. This structure agrees well with the assumption that the correlated signals in the spike counts are embedded in a largely uncorrelated noise. If the neural activities from different neurons are assumed to be conditionally independent, i.e., any correlation in the spike counts are attributed only to correlation in the underlying firing rate, the covariance matrix of the normalized neural activity can be decomposed as (see [A.17]):

where, * _{N}* = [

__Ñ___{1}

__Ñ___{2}...

*] is the matrix of the normalized spike counts (Equation A.16) of all the neurons, is the corresponding matrix of normalized rate-parameters (Equation A.18), and*

__Ñ___{n}*Diag*(·) is a diagonal matrix with

*F*

_{i}^{−1}=

*E*[

*N*]/var(

_{i}*N*)

_{i}*i*= 1, ...,

*n*on the diagonal. Thus, the gradually slopping level of the percent variance carried by the PCs can be attributed to the gradually varying Fano factors

*F*

*. In contrast, the excess variance of the initial PCs above the background level reflects correlated activity, which can be attributed to common signals that contribute to rate modulations.*

_{i}### Ensemble Permanence

Under the above assumptions, the initial principal neurons define the neural ensembles that carry the common modulating signals. Thus, their identity and in particular their permanence with time reflects the dynamics of neural computation and impacts the ability to extract the relevant modulating signals (Zacksenhouse et al., 2005).

In order to evaluate the permanence of the principal neurons with time, their identity is determined using PCA on small windows of time, and compared across time. Each principal neuron is defined by a single vector *v** _{i}* in the

*n*-dimensional neural space, which describes the relative weight given to each recorded neuron. Similarly, the initial

*m*principal neurons define an

*m*-dimensional subspace. The permanence of the

*m*-dimensional subspaces defined by the initial

*m*principal neurons can be assessed using either: (1) the cosine angle between the subspaces, or (2) changes in the variance carried by these subspaces.

Specifically, let *v*_{1}(*k*) be the first principal neuron in the *k*th window, as depicted in Figure 4.12 (left panel) for the experiment analyzed in Figure 4.11. Visual inspection suggests that the same ensemble of neurons contribute to the first principal neuron along the experiment. This conclusion is quantified by the percent variance carried by *v*_{1}(*k*) in the *l* th window (top right) and the cosine angle between the first principal neurons cos(*v*_{1}(*k*), *v*_{1}(*l*)) = *v*_{1}(*k*) *v*_{1}(*l*) (bottom right).

## LINEAR REGRESSION

The regression problem of Equations 4.5 and 4.11 can be stated in vector form as:

Where, the data matrix *X* is a *K* × *J* matrix with *K* > *J*. For proper analysis, the data matrix includes a column of ones, which account for the bias term, whereas each other column is normalized for zero mean (and optionally for unit variance).

### Principal Component Analysis (PCA)

The normalized data matrix *X* can be decomposed using singular value decomposition (SVD), as (Hansen 1997):

where,*U* = [ *u** _{1}*,...

*u*

*]*

_{k}*and*

^{K×K}*V*= [

*v*

_{1},...

*v*]

_{J}*are orthonormal matrices, Σ =*

^{J×J}*diag*(σ

_{1},...σ

_{J})

*with σ*

^{K × J}_{1}≥σ

_{2}≥...σ

_{J}≥ 0, and the rank

*r*≥

*J*is the number of strictly positive singular values

*. Note that:*

_{i}Because *U*^{T}*U* = *I** _{K×K}* the correlation matrix of the data is given by:

Equation 4.17 defines the principal component analysis (PCA) of the data. The vectors *v*_{i}^{J×}^{1} are the eigenvectors of the covariance matrix of the data, with a corresponding eigenvalue σ* _{i}*. The eigenvectors are referred to as principal velocity patterns when considering high velocity profile or principal neurons when considering neural data. The principal components (PCs) of the data, are the projection of the data on the eigenvector

*v*

_{i}*i*= 1, …,

*J*using Equation 4.15, i.e.,

*p*

*=*

_{i}*X*

^{T}*v*

*=*

_{i}*U*Σ

*V*

^{T}*u*

*= σ*

_{i}

_{i}*u*

*. The PCs are referred to as the velocity PCs, in the case of velocity data, or the activity of the principal neurons in the case of neural data. The variance of each PC is determined by the corresponding eigenvalue as var(*

_{i}*p*

*) = σ*

_{i}

_{i}^{2}, and the percent variance carried by each PC is given by:

### Least-Square Solution

The least-square solution can be expressed in terms of the PCA (or SVD) as (Hansen 1997; Fierro et al., 1997):

The resulting mean square error of the regression (MSE) is given by:

Where the vector *y* was expanded in the orthonormal basis defined by the vectors *u** _{i}* as:

The LS optimal solution is highly sensitive to measurement errors or uncertainties because, from Equation 4.19, it depends on the inverse of the singular values. Small singular values can dominate the solution and magnify errors in the measurement vector y. Thus, it is common to use regularization to stabilize the solution. Two common regularization methods are (Hansen 1997; Fierro et al., 1997) (1) Truncated SVD (tSVD), and (2) Tikhonov regularization.

### Regularization Methods

The truncated-LS regression is obtained by truncating the LS regression of Equation 4.19 at *k* ≤ *r* ≤ *J*:

The resulting MSE is:

which increases as more terms in the LS regression are truncated.

Tikhonov regularization stabilizes the optimal LS solution by minimizing the combination of the MSE and the size of the regression vector:

When *L* = *I*, the regularized regression vector with a given λ^{2} is (Elden 1982):

The resulting MSE is:

Because

the MSE increases with λ^{2} and the minimum is achieved for the optimal LS solution *c*_{λ=0} = *c** _{LS}*.

By varying the Tikhonov parameter λ, it is possible to trade-off between robustness to uncertainties (large λ) and performance on training data (small λ).

## CONCLUSIONS

The neural activity during general, free reaching movements represents the spatiotemporal profile of the movement. This representation can be captured by a multi-lag regression of the neural activity on the velocity profile. However, given the significant inter-lag correlations, this representation cannot be captured well by combining single-lag analysis. Interestingly, the neural activity is represented better in terms of the multi-lag tuning curves, which reveal how the preferred direction and depth of modulation changes with the lag, rather than in terms of tuning to the Principal components of the velocity profile.

Likewise, velocity prediction involves the spatiotemporal activity across an ensemble of neurons at multiple lags. The resulting high-dimensional regression problem requires regularization techniques for stabilizing the solution in the presence of neural noise and uncertainties due to the potential contributions of other modulating signals.

A critical issue for future BMI improvements is whether the neural activity encodes other relevant signals, aside from the spatiotemporal profile of the movement velocity. To assess this issue we developed a measure, termed the *percent overall modulations*, which quantifies the percent variance that can be attributed to neural modulations under the Poisson assumption. Comparing the percent overall modulations with the percent variance that can be attributed to the velocity profile suggests that the neural activity is modulated by additional signals, whose exact nature is still under investigation.

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## APPENDIX—DERIVATION OF THE VARIANCE RELATIONSHIP

The relationship between the statistics of the spike counts and those of the underlying stochastic information process are stated in (Snyder, 1975), with a proof based on the moment generating function. Here we provide a direct proof for the variance relationship, and for the cross-covariance between the spikes counts elicited between two doubly stochastic Poisson processes.

### Variance Relationship For Doubly Stochastic Poisson Processes

Let {*N** _{t}*} be a doubly stochastic Poisson process with intensity {λ

*(*

_{t}*x*

*)}*

_{t}*,*where

*x*

*is the underlying information process. The mean and variance of the number of spikes in bins of width*

_{t}*b*, {

*N*

*}, are related to the mean and variance of the underlying stochastic Poisson parameter:*

_{b}according to:

#### Proof

By definition:

Using the method of conditioning (Snyder 1975, Equation 6.1 there)

where the expectation is with respect to the stochastic parameter Λ* _{b}*.

Substituting Equation A.3 in Equation A.2:

and

Because the expectation is with respect to Λ_{b} it can be moved outside the summation, so the first part of Equation A.4 implies that:

where *m* = *n* − 1, and the last step is based on the equality

This proves the relationship between the mean of the spike counts and the underlying rate parameter.

The second part of Equation A.4 implies that:

where *l* = *n* − 1.

Finally, the last two equations can be manipulated to derive the following relationship between the respective variances:

This completes the proof of the variance relationship stated in (A.1).

The cross-correlation and co-variance relationship between two spike trains that are generated by two doubly stochastic Poisson processes are derived next.

### Cross Variance Relationship for Doubly Stochastic Poisson Processes

Let {*N*_{1}} and {*N*_{2}} be the number of spikes in bins of width b that were elicited by two doubly stochastic Poisson processes with underlying stochastic Poisson parameters Λ_{1} and Λ_{2} (the index b, indicating the bin-width is omitted for simplicity, but is the same for both processes). If the two processes are conditionally independent the cross-covariance between the spike counts is

#### Proof

By definition:

Using the method of conditioning

Where the expectation is with respect to stochastic parameters Λ_{1} and Λ_{2}. Conditional independence implies that given the two rate parameter, the two spike counts are independent, i.e., any correlation between the two spike counts is generated only by correlation between the underlying rate parameters. Hence:

Substituting Equation A.12 in Equation A.10, and making the change of indices *m*_{1} = *n*_{1}−1 and *m*_{1} = *n*_{1} − 1, result in:

Hence (using also Equation A.1),

In summary, Equations A.14 and A.8 imply that:

The spike counts can be normalized to zero-mean and unit variance according to:

Thus, the covariance of the normalized spike counts is:

Where the normalized rate-parameter is:

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