Recasting brain-machine interface design from a physical control system perspective
Recasting brain-machine interface design from a physical control system perspective
Yin Zhang 0 1 2
Steven M. Chase 0 1 2
Action Editor: Simon R Schultz
0 Center for the Neural Basis of Cognition, Carnegie Mellon University , Pittsburgh, PA , USA
1 Robotics Institute, Carnegie Mellon University , Pittsburgh, PA , USA
2 Department of Biomedical Engineering, Carnegie Mellon University , Pittsburgh, PA , USA
With the goal of improving the quality of life for people suffering from various motor control disorders, brain-machine interfaces provide direct neural control of prosthetic devices by translating neural signals into control signals. These systems act by reading motor intent signals directly from the brain and using them to control, for example, the movement of a cursor on a computer screen. Over the past two decades, much attention has been devoted to the decoding problem: how should recorded neural activity be translated into the movement of the cursor? Most approaches have focused on this problem from an estimation standpoint, i.e., decoders are designed to return the best estimate of motor intent possible, under various sets of assumptions about how the recorded neural signals represent motor intent. Here we recast the decoder design problem from a physical control system perspective, and investigate how various classes of decoders lead to different types of physical systems for the subject to control. This framework leads to new interpretations of why certain types of decoders have been shown to perform better than others. These results have implications for understanding how motor neurons are recruited to perform various tasks, and may lend insight into the brain's ability to conceptualize artificial systems.
Brain-machine interfaces; Decoding algorithm; Physical control system
Brain-machine interfaces (BMIs) are a powerful class of
assistive devices that may one day restore movement
ability to paralyzed individuals (Schwartz et al. 2006). These
devices act by creating a direct mapping between recorded
neural activity and the movement of an external actuator,
like a computer cursor or a robotic arm (Fig. 1) (Chapin
2004; Serruya et al. 2002; Taylor et al. 2002; Carmena et al.
2003; Musallam et al. 2004). Early clinical trials with
intracortical BMIs, which use as their control signal the recorded
activity of populations of single neurons, have recently
shown that paralyzed individuals can effectively control
computer cursors (Hochberg et al. 2006) and robotic arms of
varying complexity (Hochberg et al. 2012; Collinger et al.
2013; Aflalo et al. 2015). However, much work yet needs
to be done to give subjects control over artificial limbs that
might rival control of the natural limb (Gilja et al. 2012).
The design of BMI control algorithms that might enable
stable, robust, and naturalistic control is an active area of
research (Gilja et al. 2011).
A BMI decoding algorithm specifies how recorded
signals (like recordings from intracortical multielectrode
arrays) get translated into movement of the prosthesis.
Fig. 1 Schematic of a BMI as a feedback control system. The major
components of a feedback control system (namely, the controller,
control signals, plant, and feedback) are laid out on top of a typical BMI
cursor control schematic, where the brain is identified as the
controller, the control signals are neural activity (often tapped out of
primary motor cortex), the plant is the combination of the BMI decoder
and the cursor, and feedback is accomplished by watching the cursor
Currently, nearly all BMI decoding algorithms are designed
from an estimation standpoint: it is assumed that neurons
are tuned to different intended movements, and recorded
firing rates are treated as noisy observations of that underlying
motor intent. Thus, differences in BMI decoding
algorithms are typically interpreted to result from the different
assumptions that they make about how neurons represent
motor intent. Algorithms that fall into this class include
linear estimators such as the population vector algorithm
(PVA) (Georgopoulos et al. 1986; Georgopoulos et al. 1988;
Velliste et al. 2008) and optimal linear estimator (OLE)
(Kass et al. 2005; Salinas and Abbott 1994; Collinger et al.
2013), state-space models such as the Kalman filter (Wu
et al. 2006; Hochberg et al. 2012), Laplace-Gaussian
filter (Koyama et al. 2010a; Koyama et al. 2010a), and the
unscented Kalman filter (Li et al. 2009), as well as a host of
related variants (eg, particle filter (Brockwell et al. 2004),
SSKF (Malik et al. 2011), VBR (Li et al. 2011),
ReFITKF (Gilja et al. 2012), PT (Zhang Y and Chase SM 2013),
SDKF (Golub et al. 2014), and others). Differences in the
performance of various algorithms are typically assumed to
relate to how well the algorithms’ assumptions about motor
intent encoding match the true underlying encoding of the
However, another way to interpret BMI system design is
from a control system perspective. Each prosthetic effector,
in conjunction with its decoding algorithm, acts as a control
system that the subject needs to learn how to use through
trial-and-error. Robotic arms are, by definition, physical
control systems, while brain-controlled computer cursors
may or may not be programmed to follow physical laws.
When interpreted this way, differences in the performance
of two algorithms might instead stem from differences in the
control systems themselves: some control systems may be
more usable than others, due to their physical characteristics
or ease of conceptualization. Here we review some common
BMI cursor decoding algorithms and derive the physical
systems with which they correspond. We then re-interpret
findings from the literature on which decoding algorithms
work best in light of the physical systems that they
represent. Intriguingly, when interpreted in this way, the literature
suggests that BMI systems that follow physical laws are
more usable than those that do not. Further, in on-line
control it appears that BMI systems that reduce to equivalent
physical forms tend to be equally-well controlled. These
results have implications not only for BMI design, but may
also shed light on the brain’s ability to conceptualize motor
effectors of varying forms.
2 Linear physical systems with control signals
Before proposing our view of BMI design from a control
system perspective, we first look at a general control system,
as shown in Fig. 1. In this system, control signals generated
by the controller (the brain) are used to drive the plant (the
decoder and cursor). Feedback about the system is observed
by the sensors (the eyes) and used by the controller to
generate new control signals. If it is a physical system, the system
dynamics, which dictate the plant’s evolution as driven by
the control signal, should obey physical laws. Formally, the
dynamics for a control system can be represented as
d x/d t = f (x, u, t ),
where x ∈ Rd is the plant’s state in d dimensional space, u ∈
Rn is the control signal and t is the current time. To make
the control system a legitimate physical system, f should
obey physical laws, i.e., velocity should be the derivative of
position, acceleration should be the derivative of velocity,
For computational simplicity, a linear discretized
approximation of Eq. (1) is typically used in practice. To do that,
continuous time is discretized into a series of steps with
the step size as a small time interval Δ (Δ = 1/60s for
example), and the system dynamics at the t -th time step is
xt+1 = At xt + Bt ut .
Here, At ∈ Rd×d is a matrix that dictates the internal
physics of the control system, and Bt ∈ Rd×n is a matrix
that dictates how the control system responds to its control
If we are studying a physical system driving a
prosthetic effector’s movement, then x is often the kinematics of
the prosthetic effector, which could include the position, p,
the velocity, v, and potentially higher-order terms as well.
For example, the 1st order linear discretized approximation,
where the state is the position xt = pt , has the form
pt+1 = At pt + Bt ut .
To ensure that position is the integral of velocity, we need
to have the following relationship:
Note that the control signal affects the next velocity, only.
To gain some insight into the meaning of these control
system parameters, consider a simple 2nd order physical
system (Fig. 2). This system consists of a point mass, mp,
attached to a viscous damper and an elastic spring
connected in parallel. The forces operating on the point mass
come from external sources, Fex , as well as internal from the
spring and damper. The force from the spring is Fs = −kp
(where k is the spring constant and p is the position of the
point mass, measured relative to the equillibrium position
of the spring), and the force from the damper is Fd = −ηv
(where η is the damping coefficient and v is the velocity of
the point mass). Because these elements act in parallel, these
two forces sum at the point mass, to create a total internal
force Fin, where
Together with the external force the acceleration of the point
mass may be derived from Newton’s laws of motion:
a = Ftot /mp = (Fin + Fex )/mp = (−kp − ηv + Fex )/mp.
vt+1 = vt + Δat = vt + Δ(−kpt − ηvt + Fex,t )/mp. (7)
and the updated position is given as
Fex = u
Fig. 2 A simple 2nd order physical control system. Here, a point mass
m is hooked up to a parallel combination of a spring (with spring
constant k) and a damper (with damping coefficient η). This
configuration corresponds to the well-known Kelvin-Voigt model from material
Putting this all together into matrix form, the discretized
control system for the Kelvin-Voigt model in Fig. 2 is seen
Comparing Eq. (9) with the general 2nd order physical
control system presented in Eq. (4) reveals that the external
force Fex,t plays the role of the control signal ut , the
springlike elastic effects account for αt , and the viscous damping
effects account for βt .
3 BMI systems from a physical control systems perspective
In this section we re-interpret some fairly common
decoding algorithms, including the PVA, OLE and Kalman filter,
in terms of the physical control systems to which they
correspond. Figure 1 casts the general BMI system into the
control system framework. Here the motor cortex drives
the BMI system by generating a proper set of neural
activation patterns, which are the end result of a sequence of
brain computations that take both visual feedback and goal
information into account. Visual feedback of the prosthetic
effector’s movement is used to correct future movement.
Therefore, BMI system design is essentially a control
system design problem where one attempts to construct a BMI
system that is as usable as possible. Formally, the dynamics
of a BMI system can be represented as
d x/d t = g(x, y, t ).
where y ∈ Rn is the recorded firing rates from a population
of n neurons. The only difference between Eqs. (10) and (1)
is that the control signal u takes the form of neuronal firing
When discussing the physical implementation of
different BMI decoders, it will be necessary to distinguish the
following three variables: motor intent, estimated motor
intent, and implemented movement. Note that here and
elsewhere in this document, we use the star notation to refer
to intended movements (e.g., the intended velocity will be
denoted as v∗). Estimates of those intents will be indicated
by an overhead carat (e.g., the estimated velocity will be
denoted vˆ). Finally, the movement that is implemented by
the prosthesis will come without any notation (e.g., the
implemented velocity will be denoted as v). Please refer to
Table 1 for all the notations.
Table 1 Notations
3.1 Linear estimators
neurons’ observed firing rates
the intended kinematics (position, velocity)
the estimated kinematics (position, velocity)
the implemented kinematics (position, velocity)
We start with one of the earliest decoding algorithms,
the population vector algorithm (PVA), first introduced by
Georgopoulos and colleagues in 1986 (Georgopoulos et al.
1986). The central assumption of the PVA is that neurons
are tuned to the direction of desired movement. Formally,
with this assumption the firing rate of the neuron, y, can be
where b0, bx , and by are linear regression coefficients,
(dx∗, dy∗)T is a unit vector that points in the direction of the
intended movement, and the noise ε is assumed to be
Gaussian. For simplicity, we have written this encoding model in
two dimensions, though it has been found to generalize to
three dimensions as well (Georgopoulos et al. 1986). This
linear regression reduces to the well known “cosine-tuning”
function (Georgopoulos et al. 1982).
If we allow m to be the magnitude of the regression
coefficient vector (bx , by )T (also known as the “modulation
depth”), and we denote the “preferred direction” of the
neuron as d = (bx , by )T /m, we can write down the decoding
equation for the PVA as:
Here, i indexes each of the n recorded neurons and ks is
a constant that scales the unitless decoded direction into
a velocity (Chase et al. 2012). The general interpretation
of this equation is that each neuron “pushes” the cursor in
its preferred direction, with the amount of the push being
proportional to its firing rate. The sum of all of these
pushing vectors is decoded as the population vector, which gets
implemented in the BMI system. In this document, for
simplicity, we assume the firing rates are alway centralized so
we replace (yi − b0,i ) with yi in the following sections. We
can rewrite Eq. (12) in matrix form by gathering the
preferred directions into a single matrix D of size 2 × n, where
each column corresponds to the preferred direction of a
single neuron, and gathering the modulation depths mi into a
single diagonal matrix M:
vˆ = (ks /n)DM−1y.
To implement the PVA estimate of velocity into an actual
device, the velocity of the prosthesis is set equal to its
estimate, i.e., vt = vˆt . The implemented position is then set
equal to the integral of these velocity commands, pt+1 =
pt + Δvt . Therefore, to represent the decoding from the
physical system perspective, we have
PVA physical system: pt+1 = pt + Δ(ks /n)DM−1yt , (14)
which is a special case of a 1st order linear physical control
model (3), where At = I and Bt = Δ(ks /n)DM−1.
The PVA is a biologically-inspired algorithm. In practice,
however, it has been shown to return biased estimates of
motor intent when the preferred directions of the recorded
neurons are not uniformly distributed (Salinas and Abbott
1994; Kass et al. 2005; Chase et al. 2009). To compensate
for this bias, the optimal linear estimator (OLE) has been
proposed. As the name implies, the OLE computes the
optimal linear estimate according to square errors. The encoding
model is written as
and the decoding model is
vˆ = (BT B)−1BT y,
We can see that if the preferred directions of recorded
neurons are uniformly distributed, then BT B ∝ I and OLE
is equivalent to PVA. Similar to PVA, the implemented
velocity is also set equal to the estimated velocity in the
decoding model of OLE and the position of the prosthesis
is derived by integrating velocity. Thus, the physical system
corresponding to OLE decoding is
OLE physical system: pt+1 = pt + Δ(BT B)−1BT yt . (17)
This is again another special case of a 1st order linear
physical control system (Table 2 and 3).
Both the PVA and the OLE correspond to first-order
physical control systems, albeit with slightly different
mappings from neural firing to cursor movement. Experimental
results from (Salinas and Abbott 1994; Kass et al. 2005;
Chase et al. 2009) have shown that the OLE overcomes that
bias. From an estimation standpoint, the OLE should be a
better decoding algorithm than the PVA.
However, Chase and colleagues (Chase et al. 2009) also
demonstrated that the PVA and the OLE perform
equivalently on-line: subjects were just as adept at controlling the
PVA as they were at controlling the OLE, despite the fact
that neural activity was mapped to different cursor
movements under the two algorithms. One possible interpretation
of this is that subjects learn the mapping from neural
activity to decoder, be it a biased decoder like the PVA or an
unbiased decoder like the OLE. The difference between the
mapping from neural activity to cursor movement produced
1st order physical system
2nd order physical system
Table 2 BMI decoders under physical system perspective
Not a simple physical system
PVKF position implementation
under these two decoders is akin to a visuomotor
distortion, and visuomotor distortions are learned very quickly
(Krakauer et al. 2000; Paz et al. 2005; Wu and Smith
2013). From this standpoint, the neurons are rapidly
changing their activity to provide appropriate control signals to
the device. Once learning is accomplished, both the OLE
and the PVA give the subject a 1st order linear physical
control system to control, and there appears to be no
significant difference between the usability of these systems. This
learning process is somewhat intuitive. Experiments
demonstrate subjects can even learn the shuffled decoder with no
predictive power after several days of practice (Ganguly and
Carmena 2009; 2010).
3.2 Linear state-space decoders
A problem with the linear estimators described in the
previous section is smoothing: when implemented on small
time bins Δ, the movement estimates can be quite noisy. To
compensate for this it is common to smooth the firing rate
estimates (Koyama et al. 2010a). Naturally, any smoothing
also affects the physical system.
Linear state space models handle smoothing in a more
elegant manner by applying a smooth prior to the evolution
−Kv,t Bp Av − Kv,t H
of the intended kinematics. This prior takes the form of a
linear dynamics equation:
where ωt ∼ N (0, R) is assumed to be zero mean
Gaussian noise. The observations (recorded spike counts) are
assumed to be linearly related to the kinematics:
where again the noise in the fit is assumed to be Gaussian,
εt ∼ N (0, Q).
In this implementation, the decoding problem is a
Bayesian inference problem where the goal is to estimate
the a posteriori probability of the intended kinematics when
given the recorded spike counts. The solution in this case
is the well-known Kalman filter (Kalman 1960), first
introduced for BMI use by Wu et al. (Wu et al. 2003). The
Kalman filter provides an efficient recursive algorithm to
compute the posterior probability p(xt∗|y1,...,t ). The optimal
estimate xˆ of the intended kinematics x∗ is the mean of this
distribution, which can be estimated through the following
set of recursive equations:
Kt = (A ˆ t−1AT +R)BT(B(A ˆ t−1AT +R)BT +Q)−1, (20)
Table 3 BMI decoders comparison
Scenario (decoder found to perform best in that scenario)
off-line trajectory reconstruction (OLE)
on-line closed loop control (equivalent)
off-line trajectory reconstruction (VKF)
PVKF-position vs. VKF
PVKF-velocity vs. VKF
xˆt = (A − Kt BA)xˆt−1 + Kt yt ,
ˆ t = (I − Kt B)(A ˆ t−1AT + R).
Here Kt is the familiar Kalman gain which computes the
optimal mixing between reliance on information from the
prior and information from the observations according to
the noise assumed to be in each. Σˆ t is the estimate of the
covariance of p(xt∗|y1,...,t ).
To use the Kalman filter, an initial covariance matrix Σˆ 0
and kinematics xˆ0 must be given. In practice, it is common
to center the prosthesis so all terms in xˆ0 = 0 with certainty
Σˆ 0 = 0. The Kalman gain is time dependent. However, it
tends to converge to a stable value within a few timesteps,
independent of the observations y (Chui and Chen 2009;
Malik et al. 2011). Another common practice is to initialize
the system with xˆ0 and Σˆ 0 as the matrix that ensures that Kt
is stable (Dethier et al. 2011; Sadtler et al. 2014).
The relationship between the kinematics xt that are
actually implemented and the estimated kinematics xˆt depends
on exactly how the Kalman filter is implemented. Here
we introduce two popular implementations, the velocity
Kalman filter (VKF) (Kim et al. 2008; Hochberg et al. 2012)
and the position-velocity Kalman filter (PVKF) (Wu et al.
2006; Homer et al. 2013; Gowda et al. 2014).
3.2.1 Velocity Kalman filter
The VKF assumes that neurons are tuned to the intended
velocity. Thus, the state evolution equation is
and the observation equation is
From Eq. (21), we have the estimated intended velocity as
vˆt = (A − Kt BA)vˆt−1 + Kt yt .
The implemented velocity vt in this case is set equal to
vˆt and the position is the integral of the velocity (Kim
et al. 2008; Hochberg et al. 2012). In matrix form, the
implemented movement can be written as
Comparing with Eq. (4), we can see that the VKF is a 2nd
order linear physical control system where the system state
includes position and velocity, xt = (pt , vt )T , with an
elastic term, αt , that is equal to zero and a viscous term, βt =
A−Kt BA. It is interesting to note the parallel between force
and velocity representations that emerge in this
implementation of the Kalman filter. Even though the VKF makes the
assumption that neurons are driven by intended velocities,
yt = (Bp, Bv)
they play the role in Eq. (26) of providing a force input to
To our knowledge, nobody has directly compared the
VKF to either the OLE or the PVA in on-line control.
However, Koyama and colleagues demonstrated that a variant of
the VKF called the Laplace-Gaussian filter (LGF)
outperformed the PVA and OLE on-line (Koyama et al. 2010a).
The LGF and the VKF are both state-space decoders, and
differ in only two main respects: the LGF fits an observation
model that assumes Poisson noise statistics and log-linear
tuning to intended velocity, whereas the VKF assumes linear
Gaussian tuning of neurons to intended velocity.
Importantly, this implies that the physical control system
representing the VKF and the LGF will be of the same form:
second order with no elastic terms.
Koyama and colleagues performed simulations and
offline trajectory reconstructions to determine that the key
factor that allowed the LGF to outperform the PVA and OLE
was its state-space formulation: they found no significant
differences in the performance of the LGF relative to the
VKF. We therefore take this as indirect evidence that the
VKF would outperform the OLE and the PVA on-line. Why
should this be the case? From an estimation standpoint, the
interpretation would be that intended velocities really do
evolve smoothly over time as implied by Eq. (23), and so
incorporating the fact enables better estimates of the
velocity intent. However another interpretation is that the VKF
and OLE are fundamentally different control systems, and
2nd order physical control systems may simply be easier to
control than 1st order physical control systems.
3.2.2 Position velocity Kalman filter
Another widely used Kalman filter model is the PVKF. In
contrast to the VKF, the PVKF assumes that neurons are tuned
to both the intended position and intended velocity. Thus,
the state is xt∗ = (pt∗, vt∗)T and to encourage the state
evolution model to obey physical laws, it is typically set to be
From Eq. (20) we can compute Kt . Dissociating Kt into
two parts corresponding to position and velocity as Kt =
(Kp,t , Kv,t )T , we can write the estimated intended position
and velocity as
The PVKF observation model is
where H = ΔBp + Bv Av
In the PVKF, the estimated position is not, in general,
equal to the integral of the estimated velocity. Even though
the state evolution equation (27) biases estimates of
position and velocity to obey this rule, it is not a hard constraint:
a compromise between position and velocity will be
estimated that best explains the observed spike rates. This then
leaves one with a choice when trying to implement the
PVKF, since both the position and velocity estimates
cannot be simultaneously implemented. One common method
is to use the estimated position as the implemented
position, and to allow the implemented velocity to evolve as
vt = (pˆ t+1 − pˆ t )/Δ (Wu et al. 2006):
PVKF, position implementation:
(Not a simple physical control system)
I − Kp,t Bp ΔI − Kp,t H
−Kv,t Bp Av − Kv,t H
With this implementation, the estimated velocity, vˆt ,
becomes a latent variable that keeps the system from
having a simple physical control system correlate. Experimental
results from (Kim et al. 2008) show that when the PVKF is
implemented in this way, its performance is inferior to the
performance of the velocity-only Kalman filter. This fact is
hard to rationalize from the estimation viewpoint, since the
encoding model assumed by the PVKF typically fits the
firing rates better than the encoding model assumed by the
VKF. From the control system viewpoint, however, a
possible explanation of this result is that the PVKF with position
implementation is not a simple physical control system, and
is therefore difficult to use.
There are other ways to implement the PVKF. One way
is to use the estimated velocity as the implemented
velocity and treat the estimated position pˆ t as the hidden variable.
Another is to use a linear combination of estimated
velocity and estimated position as the implemented velocity. This
latter method was shown in (Homer et al. 2013) to work
better than the position-implementation of the PVKF. However,
none of these versions give the subject a simple physical
system without hidden states to learn to control.
There is one implementation of the PVKF, however, that
does correspond to a simple physical control system with
no hidden states. To do this, the implemented velocity is
made equal to the estimated velocity, and the estimated
position becomes the integral of the estimated velocity:
PVKF, velocity implementation:
(Physical control system)
As it turns out, this implementation of the PVKF has
been used by Gilja and colleagues to achieve the best
BMI control demonstrated to date (Gilja et al. 2012). Their
implementation of this equation was one of two design
innovations of their ReFIT-KF algorithm, the other being a new
method for re-calibrating the device from on-line training
data. Although they derived (31) in a different way, using a
causal-intervention step that forces the variance of the
position estimate to go to zero, the net effect is the same, and
leads to a second order physical system with both elastic and
The velocity-implementation of the PVKF has been
shown to handily outperform the VKF (Gilja et al. 2012).
One interpretation of why this is true is that the causal
intervention step better captures the fact that subjects know
the position of the cursor from sensory feedback, so there
should be no uncertainty about it. While this clearly cannot
be exactly true, due to sensory feedback delays and sensory
noise (Golub et al. 2013), it may be true enough to allow for
better control. Another interpretation might be that neurons
are driven by a non-volitional, positional signal
representing the real position of the cursor, and this equation allows
that “nuisance variable” term to be removed. However, yet
another interpretation might be that this implementation of
the PVKF results in a simple, 2nd order physical control
system that still allows for neurons to modulate as a function of
position. It may further be the case that 2nd order physical
control systems that include a certain elastic component (a
non-zero αt in Eq. (4)) are more usable than those systems,
like the VKF, that do not have that component.
The optimal way to implement a BMI decoding algorithm
is an important question relevant to clinical deployment of
neural prostheses. Here we recast this problem from the
perspective of control system design, and derive the physical
control systems corresponding to various types of decoders
commonly used in BMI cursor control. This process enables
new insights into BMI design, and suggests novel
explanations about why some decoders have been shown to
perform better than others. In particular, the literature
suggests that: 1) 2nd order physical systems tend to be more
usable than 1st order physical systems, 2) decoders that
cannot be expressed as simple physical control systems do not
appear to work as well as those that can be expressed this
way, and 3) a 2nd order control system with elastic terms
seems to work better than one without.
Recent work has highlighted the utility of approaching
BMI design as a separate problem from inferring
natural behavior (Tillery and Taylor 2004; Chase and Schwartz
2010). Marathe and Taylor (Marathe and Taylor 2011)
studied the effect of mapping one control parameter (e.g.,
position, velocity, or goal) to the control of another. They
found that the optimal mapping was not necessarily
one-toone, but rather changed as a function of different types of
decoding noise. Gowda and colleagues (Gowda et al. 2014)
have presented a thorough investigation of the dynamical
systems properties of the PVKF. They found that certain
implementations of the decoder could create workspace
attractor points that might be detrimental to BMI control.
These studies point out the gains that may be realized when
BMI control is not constrained to reflect the neural
encoding of natural arm dynamics, and emphasize the importance
of the physical control system perspective when interpreting
4.1 Suggestions for new BMI systems
Our analyses suggest a number of new approaches to BMI
decoding algorithm design that might prove fruitful. Of all
the decoding algorithms we reviewed, none went beyond
2nd order. Given that 2nd order systems appear more usable
than 1st order systems, it is interesting to speculate as to
whether a 3rd or 4th order system would be even easier to
control. These higher-order systems may actually be a closer
match to the human arm: in (Liu and Todorov 2007), Liu and
colleagues model the arm as a 3rd order linear physical
system and are able to capture many of the emergent features
of natural reaching movements. There have been instances
in the literature that have included acceleration and higher
order terms in their decoding algorithms. For example, Wu
and colleagues compared the performance of a Kalman
filter decoder with up to 6th order terms, and found that the
3rd order model consisting of position, velocity, and
acceleration terms provided the best performance in their off-line
trajectory reconstruction (Wu et al. 2006). However, they
used the position-implementation of their decoder, which
we have already demonstrated does not correspond to a
simple physical system beyond 1st order. It would be interesting
to test how physical implementations of higher order control
systems might perform on-line.
Another fruitful approach might be the design of
statedependent control systems. The PVA and the OLE are
both static systems, i.e., the physical control parameters
do not vary with time. The Kalman filter technically has
time varying parameters, but in practice the Kalman gain
converges to a constant within a few timesteps, and some
researchers even initialize it at the converged-values to keep
it time-invariant (Malik et al. 2011; Sadtler et al. 2014).
On the other hand, state-dependent control is also studied
to deal with the situations where static system is no longer
appropriate. One such situation is the instability of
tuning curves between sessions (Rokni et al. 2007; Chestek
et al. 2009). To compensate for such instability, different
adaptive decoding algorithms are proposed where
decoding parameters are updated iteratively over time (Li et al.
2011; Shpigelman et al. 2009; Zhang Y and Chase SM
2013; Suminski et al. 2013; Kowalski et al. 2013;
Orsborn et al. 2014). Another kind of state-dependent control
is used to assist the subject during training with the BMI
system, where the assistance is adjusted according to the
subject’s performance (Velliste et al. 2008). Other
statedependent control algorithms include (Shanechi et al. 2013),
where in order to simultaneously estimate movement
trajectory or target intent, the decoding parameters are adjusted
as targets are approached. Such state-dependent control was
the goal behind the recently-developed speed dampening
Kalman filter (Golub et al. 2014), which effectively
manipulated the viscous damping forces as a function of trajectory
curvature to allow for more stable transitions from
moving to stopping. The physical control system viewpoint we
espouse here may be a way of integrating these approaches
into a simple, unified framework for robust prosthetic
Finally, we should note that we have focused exclusively
in this review on kinematic BMI decoders of the type that
are commonly used to drive cursors on computer screens.
Another class of decoders attempts to extract kinetic
information (forces and torques) from recorded neural activity
for direct control of force output (Fagg et al. 2009;
Nazarpour et al. 2012; Nishimura et al. 2013; Chhatbar and
Francis 2013; Oby et al. 2013). It is unclear at present
how to best integrate these two approaches to BMI design.
One intriguing idea is that the brain represents impedances
(relationships between kinetics and kinematics) rather than
desired forces or kinematics per se (Hogan 1985; Tin and
Poon 2005; Hogan and Sternad 2012). An interesting, active
direction of research is to design decoders that seemlessly
transition between free movement and object interaction. It
is possible that decoding impedance control signals directly
from the brain would enable this transition.
4.2 Do details of the control signal mapping matter?
We have focused primarily on the overall form of the
physical control system represented by different decoders: e.g.,
whether it is 1st order or 2nd order and whether it contains
both viscous and elastic elements, etc. We have not focused
on the particular values of those parameters as much, except
to point out that the PVA and OLE, which have different
mappings between firing rates and cursor movement,
perform similarly on-line (Chase et al. 2009). Do the details of
the control system parameters matter at all?
The answer is certainly yes. Sadtler and colleagues have
recently demonstrated that some mappings between
neural activity and cursor movement are easily learned, while
others are very difficult to learn (Sadtler et al. 2014). In
that study they discovered that subjects could easily learn
to map an existing pattern of neural activity to an arbitrary
movement; what was difficult was creating new
correlation patterns in the neurons themselves. This suggests that,
provided the mapping from neural activity to cursor
movement preserves the natural correlations within the neural
population, two physical systems of the same form may be
Also, while Sadtler and colleagues noted substantial
improvement in performance with short amounts of practice
on novel ‘within-manifold’ (correlation preserving)
mappings, the performance under those mappings was still
substantially worse than performance with the
initiallyestimated natural tuning curves. Further, Gilja and
colleagues (Gilja et al. 2012) have noted a significant
improvement in control if tuning curves are re-estimated from
on-line training data, which demonstrates different
mappings between firing rates and cursor movement can lead to
different on-line performance.
Under the estimation-framework, performance will be
best with mappings from neural activity to cursor movement
that best capture the natural neural tuning. Under the control
framework, performance will be best with those mappings
that most accurately capture the volitionally usable
correlation structures within the neural population. Of course, all of
this ignores long-term learning, which may allow subjects to
become proficient even at non-intuitive mappings (Ganguly
and Carmena 2009; 2010). Further work will need to be
done to determine the precise details that determine which
mappings from neural activity to movement perform better
than others, and how learning may ultimately play a role in
sculpting their performance (Orsborn et al. 2014).
4.3 Alternate views of motor cortical recruitment
Decades of motor control studies have established that
neural activity in motor cortex correlates with various features
of movement. In BMI design, it is common to interpret
these correlations by thinking of neurons as being tuned
to the desired outcome or intended movement of an
effector. However, neural activity can be flexibly dissociated
from the effector (Schieber 2011): using operant
conditioning, individual neurons can be trained to correlate and
decorrelate from particular muscles (Fetz 1969; Fetz and
Finocchio 1971), even when spike triggered averaging of
EMG traces provide evidence of monosynaptic
connections between the neuron and the muscle (Davidson et al.
2007). Evidence of a flexible relationship between neural
activity and behavior also comes from natural movement
paradigms. Neural activity in motor cortex readily changes
during motor learning (Jarosiewicz et al. 2008; Paz et al.
2003; Wise et al. 1998; Gandolfo et al. 2000; Ganguly and
Carmena 2009; Sadtler et al. 2014). Neural activity has also
been shown to change during associative learning: neurons
in M1 will respond to the color of a target during an
associative learning task, and will often maintain that tuning
after color is no longer relevant (Zach et al. 2008). Even
context can modulate neural tuning. Hepp-Reymond and
colleagues (Hepp-Reymond et al. 1999) demonstrated that
neurons in primary motor cortex are sensitive to the context
of an isometric force task, and will change their firing to
particular force levels according to how many force targets
are presented in the task.
These studies suggest that motor cortical activity is
extremely fungible, restructuring itself in the face of new
task demands in a manner that is not at all well understood.
It is hard to reconcile this view with a static view of
tuning to intended movements, unless one assumes that these
motor intent signals can themselves be dissociated from the
motor outcome (Chase and Schwartz 2010). If this is the
case, there may be only subtle differences among the
viewpoints that neurons tune to flexible motor intent signals, that
neurons act as control signals to drive an effective behavior,
or that populations of neurons act as a flexible pattern
generator on which movements can be built (Shenoy et al. 2013).
In this review, we mainly wish to highlight the importance of
these multiple viewpoints when attempting to interpret BMI
performance and, ultimately, design the optimal decoding
4.4 Relationships to embodiment, internal models,
and natural motor control
Natural motor control is fraught with computational
difficulties, not least of which is the necessity to compensate
for noisy, delayed sensory feedback. To generate fast,
dexterous movements, it is necessary to compensate for these
sensory delays. It is widely believed that we do this with
the aid of internal models that allow us to predict, in real
time, the outcomes of our motor commands before sensory
feedback becomes available (Crapse and Sommer 2008;
Shadmehr et al. 2010). These internal (forward) models are
thought to take as input efference copies of our motor
commands, and use them to predict the sensory consequences,
such as the new arm or eye position, that result from those
commands (Sommer and Wurtz 2002; Wolpert et al. 1995).
Essentially, these models embody our internal conception of
the physics of our limbs and how they respond to our motor
commands. These predicted locations can then be used as
the basis for planning the next movement before real
sensory feedback becomes available, allowing for faster motor
Internal models may also be used in BMI control. Motor
commands of subjects using a BMI to control a computer
cursor in a center-out movement task are more
appropriate to the real time position of the cursor than the last
sensory feedback position, indicating that subjects
compensate for sensory feedback delays while using a BMI (Golub
et al. 2012). Further, differences between a subject’s internal
model of the decoder and the actual cursor dynamics may
explain errors in on-line control (Golub et al. 2013).
It is intriguing to speculate on the utility of internal
models in BMI control. Internal models are thought to be a
key component in motor adaptation (Shadmehr et al. 2010;
Kawato 1999), and thus subjects who build internal models
of BMIs may be able to take advantage of all of the
computational motor adaptation machinery that natural motor
control relies upon. A reliable internal model of the BMI
device may also be the key to embodiment, when the device
feels like a natural extension of one’s body (Schwartz et al.
2006; Giummarra et al. 2008). However, can subjects build
internal models of any BMI device? While the answer to this
question is well beyond the scope of this review, we posit
that it might be easier to build internal models of physical
systems than it is to build internal models of non-physical
systems, simply because all of the internal models we build
in the context of natural reaching are, by definition,
models of physical systems. It could be that the reason decoding
algorithms corresponding to physical systems appear to be
more useable than those that do not is because they are more
readily conceptualized with an internal model.
Acknowledgments This work was supported by the Pennsylvania
Department of Health Research Formula Grant SAP #4100057653
under the Commonwealth Universal Research Enhancement program.
We thank A. Batista, W. Bishop, M. Golub, S. Kennedy, P. Lund, E.
Oby, S. Perel, K. Quick, P. Sadtler and B. Yu for insightful comments
on a preliminary version of this paper.
These authors declare no conflict of interests.
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