There is an interesting section on Approaches for BCI Learning in the review Brain computer interface learning for systems based on electrocorticography and intracortical microelectrode arrays. Specifically:

Since cortical stimulation can modulate cortical activity patterns (Hummel and Cohen, 2006; Harvey and Nudo, 2007), it is conceivable that cortical stimulation may be able to replace or supplement repetitive behavior training to induce changes in cortical activity and accelerate BCI learning (Soekadar et al., 2014). While this approach has not been well investigated for BCI learning, previous studies about neuroplasticity (Gage et al., 2005; Jackson et al., 2006) and rehabilitation using neurostimulation (Ziemann et al., 2002; Hummel et al., 2005; Hummel and Cohen, 2006; Harvey and Nudo, 2007; Perez and Cohen, 2009; Plow et al., 2009; Reis et al., 2009) can shed some light on the feasibility of this approach. At the macroscopic level, cortical areas can be stimulated non-invasively using transcranial magnetic or current stimulations. In the context of stroke rehabilitation, it has been suggested that such stimulation can enhance motor cortical excitability and change cortical connectivity (Hummel et al., 2005; Hummel and Cohen, 2006; Perez and Cohen, 2009). [...] A recent pilot study has shown that transcranial direct current stimulation induces event-related desynchronization associated with sensorimotor rhythm (Wei et al., 2013). This event-related desynchronization, along with motor imagery, was used to improve the performance of an EEG based BCI.

Nothing interesting there, there have been quiet some evidence that anodal tDCS has positive effects on motor and cognitive skill acquisition. I particularly like Soekadar 2014: tDCS was used to help with the subjects to train to modulate sensorimotor rhythms (SMR, 8-15Hz). They hypothesized that M1 had a causal link to modulate SMR, so anodal tDCS was applied there. They found anodal stimulation resulted in better performance than sham and cathodal stimulation. I will not comment on if this experiment alone establishes its conclusion that "M1 is a common substrate for acquisition of physical motor skills and learning to control brain oscillatory activity", but it certainly serves as evidence that tDCS may help in BMI control acquisition.

At the microscopic level, based on the concept of Hebbian or associative learning, motor cortical reorganization can be induced by coupling action potentials of one motor cortical neuron with electrical stimulation impulses of another motor cortical neuron (Jackson et al., 2006; Stevenson et al., 2012). Besides electromagnetic stimulation, optogenetics is another approach to stimulate cortical tissue.

They reference the Jackson, Mavoori, and Fetz 2006 paper Long-term motor cortex plasticity induced by an electronic neural implant. In this paper, stimulation was delieverd on one electrode upon detection of action potential on another one. This was done for 17 pairs of electrodes over 8 to 9 sessions spread between 1 to 4 days. The stimulation current is just above threshold current needed to elicit a muscle response (wrist). They first measured the muscle response torque vector elicited from stimulating the recording electrode, stimulating electrode, and a control electrode. After conditioning, they showed that the response vector elicited from stimulating the recording electrode has has shifted toward the response vector of the stimulating electrode. Meanwhile, the control electrode response vector does not change significantly. This results seems consistent with the second neuron firing in sync with the first neuron. Furhter, varying the lag betweeen record and stimulation has an effect on this shift, consisten with spike-timing dependence. The authors then suggest that this can be a method to induce "artifical synapses". While the use of even cortical stimulation to selectively strengthening specific neural pathways during rehab is not a brand new idea, the ability to create artificial connections is crazy cool for BMI.

In BMI, we know exactly (exaggeration) what signals will be mapped by a decoder to movement commands for a BMI. This means if we use a random decoder assigning weights to the units recorded at the different electrodes, we can potentially apply stimulation at those electrodes according to our decoding weight matrix to enhance the BMI skill acquisition.

A more recent study from Fetz' group Paired stimulation for spike-timing-dependent plasticity in primate snesorimotor cortex uses slightly different approach to induce STDP. They first map out connections between neurons near implanted pairs of electrodes, judging from the measured LFP in response to stimulation. Neuron A is putatively pre-synaptic, Neuron B post-synaptic, and Neuron C has recurrent connections with both of them and serves as a control. During conditioning they stimulated A then followed by B. The results again measured the evoked EP for A->B, B->A, and all to C.

Results are mixed: 2 out of 15 pairs showed increased EP in A->B direction. Network changes were seen -- neurons near the implanted area not involved in paired-stimulation also showed changes in their EP. Some showed depressed EP. Conditioning with spike-triggered stimulation from the 2006 study produced a larger proportion of positive plasticity effects than paired electrical stimulation and the authors propose possible mechanisms:

1. Stimulating at site A rather than using recorded trigger spikes would have activated a larger population of more diverse cell types and consequently can recurit a broad range of plasticity mechanisms, such as anti-Hebbian effects.

2. The triggering spikes in 2006 study occured in association with normal behavior, whereas the paired stimulation was delivered in a preprogrammed manner independently of the modulation of local activity with movements or sleep spindles.

3. Most importantly, the 2006 study measured plasticity effects in terms of behavior, rather than EP.

So, looks like behavior is important and spike-triggered stimulation may result in better plasticity mechanisms, I hypothesizie.. Ideally, stimulation specificity afforded by optogenetics may be a better tool to study this effect.

And apparently someone in our group from 2010-2012 had a similar idea about using stimulation to assit in BMI skill acquisition, and the results were bad, resulting in mastering-out.. So I better put this on the back burner.

The problem with using intention estimation in BMI is that the algorithm designers need to write down an objective function. While this is easy to do for a simple task, it is unclear how and not practical to do this for a general purpose prosthesis.

Using reinforcement learning based decoder deriving error-signal directly from cortical activities would be nice and in my opinion the holy grail of BMI decoder. However, deriving the proper error signal and constructing an error decoder seems to present the same problem -- the designers have to first establish what exactly is an error condition.

A compromise is perhaps a human assisted search in the decoding space. The possible decoding space is large, but as Google+Tri Alpha demonstrated, by guiding MCMC search via human inspection of the results, good results in such complicated problems such as plasma confinement is possible.

This general approach of learning from human preferences is also becoming a hot topic recently, as getting an objective function slightly wrong may result in completely unseen consequences (much worse than arrogantly assuming a monkey's goal in an experiment is to always get a grape).

FemtoDuino Offical Site

FemtoBeacon Specs

We bought the 5-coin + dongle version. Each chip has onboard a ATMEL SAM R21E (ATSAMR21E18A), which is Arduino compatible. The coins have onbaoard precision altimeter and a 9 Axis IMU (MPU-9250). The wireless communication between the coins and the dongle can be programmed t use any available 2.4GHz RF stack. FemtoDuino implements Atmel's LwMesh stack.

The design files for the boards and example programs are in the femtobeacon repository. The repository's README includes setup instructions to program the FemtoBeacon with bareemtal C with the Atmel Software Framework. Alex the maker of FemtoBeacons personal uses and suggests using the Arduino core and bootloader instead.Discussion.

I am using the Arduino core and bootloader for my development as well.

Machine Setup

Femtoduino has a list of instructions. There is also a list of instructions on Femtoduino's github repo.

Here I compile the relevant procedures I have taken, on Ubuntu 14.04.

1. Download Arduino IDE, version 1.8.1 or higher.

2. From Arduino IDE's board manager, install Arduino SAMD Boards (32-bits ARM Cortex-M0+) by Arduino. Femtoduino recommends version 1.6.11. I have tested 1.6.7 to 1.6.12, doesn't seem to make too much of a difference.

3. Add package URL (given in ArduinoCore repo) to the Additional Board Manger URLs field in the Arduino IDE via File > Preferences (Settings Tab).

   The Stable Release URL is:

   https://downloads.femtoduino.com/ArduinoCore-atsamd21e18a/package_atsamd21e18a-release-build_index.json.

   The hourly build URL is:

   https://downloads.femtoduino.com/ArduinoCore-atsamd21e18a/package_atsamd21e18a-hourly-build_index.json.

   I have found the hourly build works without compilation errors.

4. This core is now available as a package in the Arduion IDE board manager. Use Arduino's Board Manager to install Atmel SAM D21/R21 core (ATSAMD21E18A/ATSAMR21E18A) by Femtoduino.

   At this point, with default installations, there should be a .arduino15/packages/femtoduino/hardware/samd/9.9.9-Hourly directory, which contains the Arduion core for our device. This directory should contain files in the ArduinoCore-atsamd21e18a git repo. The example files for the RF-dongle and RF-coins are in the libraries/ folder of this directory.

5. Install the FreeIMU libraries from Femtoduino's fork of FreeIMU-Updates. This is needed for the onboard MCUs to talk to the IMU chips.

   This is done by either forking or downloading FreeIMU-Updates's FemtoBeacon branch (important!). Make the libraries visible to Arduion -- two ways to do this:

   1. Copy all the folders, except the MotionDriver/ folder in FreeIMU-Updates/libraries into the Arduion libraries folder. By default, the Arduion libraries folder is under /Arduino/libraries. See Arduino library guide for more instructions.

   2. As I might make changes to FreeIMU library code, it's easier to symbolic link the libraries to Arduino's library directory. Do this with:

      cd ~/Arduino/libraries

      `cp -r --symbolic-link ~/PATH_TO/FreeIMU-Updates/libraries/* .

      Remember to delete the symbolic link to MotionDriver/ since we don't need it.

   In FreeIMU/FreeIMU.h header file (from folders copied previously), the line #define MPU9250_5611 is the only uncommented line under 3rd party boards.

   In FreeIMU/FreeIMU.h, find the following section:

   //Magnetic declination angle for iCompass
   //#define MAG_DEC 4 //+4.0 degrees for Israel
   //#define MAG_DEC -13.1603  //degrees for Flushing, NY
   //#define MAG_DEC 0.21  //degrees for Toulouse, FRANCE
   //#define MAG_DEC 13.66 // degrees for Vallejo, CA
   //#define MAG_DEC 13.616 // degrees for San Francisco, CA
   #define MAG_DEC -9.6    // degrees for Durham, CA 
   //#define MAG_DEC 0
   

   and enter the magnetic declination angle for your location. Do this by going to NOAA, enter your zip code and get the declination result. For the result, an east declination angle is positive, and west declination agnle is negative. This is needed to for magnetometer reading responsible for yaw calculations.

6. Install the FemtoDuino port of the LwMesh library. This is needed to run the wireless protocols.

   Fork or download the at86rf233 branch of FemtoDuino's fork of library-atmel-lwm. Having the incorrect branch will break compilation.

   Move all the files in that repository into the Arduino library folder as at86rf233.

7. Install the Femtoduino fork of RTCZero library, osculp32k branch. This is needed to use an external 32kHz clock onboard the chips (see compilation issue thread for discussion).

   Fork or download osculp32k branch of Femtoduino's fork of RTCZero. Move it to Arduion library folder as RTCZero.

Testing Coin and IMU

To check if the machine setup has gone successfully, in Arduion IDE, select the Board: "ATSAMR21E18A (Native USB Port)" option. If it's not available, step 1-4 of the Machine Setup was probably done incorrectly.

Select the port corresponding to the connected Coin-chip.

Open FemtoBeacon_Rf_FreeIMU_raw.ino through File > Examples > FemtoBeacon_Rf.

Compile and upload. If machine setup has gone correctly, should be able to see IMU readings in the serial monitor. Serial plotter can also visualize the outputs.

Testing Dongle

Testing the Dongle requires some modification of FemotoBeacon_Rf_MESH_IMU_Dongle.ino (can also be acessed via Files > Examples > FemtoBeacon_Rf). This program makes the Dongle listen for wireless traffic, and prints them in Serial. If there's no traffic, then nothing is done.

This is not very informative if we just want to see if the Dongle's working. So change the original handleNetworking() method:

void handleNetworking()
{
    SYS_TaskHandler();
}

to

unsigned long start = millis(); // Global variable, asdffffffffffffffffffffffffffffffffffffffffffffffffffff

void handleNetworking()
{
    SYS_TaskHandler();
    if (millis() - start > 1000) {
        Serial.print("Node #");
        Serial.print(APP_ADDRESS);
        Serial.println(" handleNetworking()");
        start = millis();
    }
}

This way, even without wireless traffic, the dongle will print out "Node #1 handleNetworking()" every second in the serial monitor.

Testing Femtobeacon Networking

Upload one Femtobeacon coin with FemtoBeacon_Rf_MESH_IMU_Coin.ino and the dongle with FemtoBeacon_Rf_MESH_IMU_Dongle.ino.

Keep the dongle plugged into your computer with Arduino IDE running, and the other coin unconnected but powered through its USB connector.

In the Serial monitor, you should be able to see outputs such as

Node #1 handleNetworking()
Node #1 handleNetworking()
Node #1 handleNetworking()
Node #1 handleNetworking()
Node #1 handleNetworking()
Node #1 receiveMessage() from Node #2 = lqi: 156  rssi: -91  data:   154.23,   16.56,   34.45
Node #1 receiveMessage() from Node #2 = lqi: 172  rssi: -91  data:   153.93,   16.89,   34.93
Node #1 receiveMessage() from Node #2 = lqi: 220  rssi: -91  data:   153.66,   17.18,   35.32
Node #1 receiveMessage() from Node #2 = lqi: 160  rssi: -91  data:   153.39,   17.43,   35.61

where Node#1 represent the dongle, Node #2 represents the coin beacon.

Calibrating IMU

Follow the official Femtoduino's instructions. Note that the calibration utility cal_gui.py is in the FreeIMU-Updates folder that was forked in machine setup step 5.

Download processing and run the cube sketch to check for results.

Before starting to collect samples from the GUI, in Arduino's Serial Monitor, send a few "q" commands (reset IMU) with some time in between or a "r" command (reset quaternion matrix) for best results. See post here

Common Errors

1. If serial port permission problems aren't setup correctly, we may see this error:

   Caused by: processing.app.SerialException: Error touching serial port '/dev/ttyACM0'..

   Solve by adding your user account to the dialout group:

   sudo usermod -a -G dialout yourUserName

2. The FreeIMU's variants of getYawPitchRoll methods doesn't actually give the yaw, pitch, and roll one may expect. In the comments for the method:

   Returns the yaw pitch and roll angles, respectively defined as the angles in radians between
   the Earth North and the IMU X axis (yaw), the Earth ground plane and the IMU X axis (pitch)
   and the Earth ground plane and the IMU Y axis.
   
   @note This is not an Euler representation: the rotations aren't consecutive rotations but only
   angles from Earth and the IMU. For Euler representation Yaw, Pitch and Roll see FreeIMU::getEuler
   

   The one that I expected is given by the getEuler() method:

   Returns the Euler angles in degrees defined with the Aerospace sequence.
   See Sebastian O.H. Madwick report "An efficient orientation filter for 
   inertial and intertial/magnetic sensor arrays" Chapter 2 Quaternion representation
   

3. After the previous step, we should have reasonable yaw, pitch, roll readings in degrees. However, the yaw reading may exhibit a huge drift/set point behavior -- while the beacon sits flat on a surface and is rotated along its z-axis, the yaw reading would initially change to some reasonable measurement, then drift back to the same value.

   As the magnetometer should be pretty stable, unless the environment has a lot of changing magnetic field, it would be due to how the sensor-fusion algorithm is updating the measurements. In FreeIMU.h, look for the lines:

   // Set filter type: 1 = Madgwick Gradient Descent, 0 - Madgwick implementation of Mahoney DCM
   // in Quaternion form, 3 = Madwick Original Paper AHRS, 4 - DCM Implementation
   #define MARG 3
   

   Out of the possible methods, only method 3 gives me yaw reading with a non-instant drift-time...

Directories

• Arduino/libraries: All the third-party libraries for lwmesh, RTCZero, FreeIMU_Updates. FreeIMU_Updates symbolic-linked to FreeIMU_Updates git repo.
• .arduino/.../femtoduino../: Arduino board-manager installed cores.

References

Compilation issue

Magnetometer Reading Explanations

Complementary Filters

Cortical ensemble adaptation to represent velocity of an artificial actuator controlled by a brain-machine interface

Context
Follow-up to the Carmena 2003, Learning to control a brain-machine interface for reaching and grasping by primates paper. In the experiments, monkey first learned to do a target-acquisition task via joystick-control (pole-control), then brain-control was used to control the cursor while the monkey was still allowed to move the pole (brain control with hand movement (BCWH)), and finally the joystick was removed (brain-control without hand movement (BCWOH)). All bin-size=100ms.

Individual cortical neurons do not have a one-to-one relationship to any single motor parameter, and their firing patterns exhibit considerable variablitiy. Precise motor control is achieved through the action of large neuronal ensembles.

The key questions addressed was: How do the neuronal representations of the cursor movement and arm movement of the recorded ensembles change between pole-control, BCWH, and BCWOH?

Method

Implants in M1, PMd, SMA, S1, and PP.

1. Tuning to velocity during pole and brain control. Constructed linear regression model to predict neuronal firing rates based on velocity:

7. Random neuron dropping: 10min of neuronal population data fit the velocity prediction model. Model then used to predict on a different 10min period. A single neuron is randomly dropped from the population, train then test. This process is repeated until no neurons remained. This entire process (dropping 1 to entire population) is repeated 100 times to yield $R$ as a function of number of neurons.

Results

1. Hand and robot velocity in pole-control vs. BCWH -- More correlated during pole-control (expected), less during BCWH. Ranges of velocity similar. In BCWH, robot moved faster than hand. Ranges of robot velocity similar between BCWH and BCWOH.

2. Neuronal tuning during different modes -- Velocity tuning defined as correlation between neuronal rate and velocity. Individual neurons exhibited diversity of tuning patterns:

   • Fig.3: Tuned to both pole-control and brain-control (shown in 3A). VTI higher before instaneous velocity measurement (IVM). VTI for differnet modes significantly different (Kruskal-Wallis ANOVA, Tukey's multiple comparison). Highest VTI during pole-control. After transitioning to brain control, this neuron retained many features of its original velocity tuning, but also became less tuned to hand movements.

   • Fig.4: Tuned to hand movements. Changes in VTI between different modes are significant. Tuning present during pole-control and BCWH, but vanished in BCWOH. Highest VTI before IVM. Observed lag-dependent PD to both hand and robot velocity. Because of the strong dependency of tuning on the presence of hand movements, this neuron is likely to have been critically involved in generation of descending motor commands.

   • Fig. 5: Tuned to BCWOH.

   Pairwise comparison of pole-control with BCWH and BCWOH (within same sessions) showed in majority of neurons, peak VTI for hand decreased after transitioning to BCWH. In BCWH, the peak VTI for robot is significantly greater than peak VTI for hand for a majority of neurons. Peak VTI during BCWOH was greater than during pole control in $38\mathrm{\%}$ of neurons.

3. Tuning patterns for ensembles

   • Fig.6: Ensemble VTI for different control mode. In majority of neurons, and also average VTI, tuning to hand movement decreased from pole-control to BCWH. VTIs for robot movement is higher pre-IVM, because only bins preceding IVM were used for online prediction, and delay of the robot/cursor introduces a lag. VTI for hand movement peaked closer to IVM than for robot movement. Average VTI similar to M1 VTI due to stronger tuning.

     Occurence of ensemble tuning to robot velocity in BCWOH is not surprising as neural activities controlled robot movement, but it's not simply a consequence of using the linear model. The shuffled test showed average VTI decreased after the shuffling procedure, indicating role of inter-neuronal correlation in cortical tuning. This corroborates Carmena 2003's finding that firing of individual neurons is correlated and increase in brain control.

   • Fig 7: Ensemble PD. Generally there was no fixed PD for each neuron and rotated with lag changes (may be due to representation of accelerative and decelerrative forces a la Sergio & Kalaska). In BCWH, PDs for both hand and robot resemble that of pole-control. Correspondence of hand PD between pole-control and BCWH is highest at IVM. For robot PD correspondence is highest 100ms before IVM, which is in agreement with average velocity VTIs. PDs during BCWOH were confined to a narrower angular range and rotated less.

4. Neuronal tuning for selected hand movements.

   • Fig 8: Comparing directional tuning to hand movements in pole-control and BCWH. Both directional tuning depth and VTI w/ respect to hand are less in BCWH. Differences are statistically significant.

5. Offline predictions of hand velocity during pole and brain control. Three prediction curves were made, 1) Using pole-control to train models, then predict pole-control; 2) Train with pole-control, predict BCWH; 3) Train with brain-control, predict BCWH.

   • Fig 9: Prediction quality: (1)>(3)>(2). This decrease in prediction accuracy applies to all lags. Suggests cortical representation of the robot ws optimized at the expense of representation of the animal's own limb.

Conclusion

1. Principle finding: Once cortical ensemble activity is switched to represent the movements of the artificial actuator, it is less representative of the movement of the animal's own limb, as evidence by how tuning to hand velocity decreases from pole-control to BCWH.

   • Neuronal tuning to robot movements may be enhanced due to increased correlation between neurons (attention leads to synchrony ${}^2$).

   • Supports evidence that cortical motor areas include cognitive signals as well as limb movements. Limb representation is highly flexible and susceptible to illusions and mental imagery (rubber arm). Neuronal mechanisms underlying adaptive properties ${}^1$ may be responsible for cortical ensemble adaptation during the BMI operation.

2. BMI design makes any modulation of neuronal activity translate into movement of the actuator. To interpret it as reflection of new representation of artificial actuators, uses the following evidence:

   • Nonrandomness of actuator movements and behavioral improvements. (Peaking of VTI pre-IVM not neccessarily sufficient because that's built into prediction algorithms).

   • Increased correlation between neurons during brain control (via Carmena 2003).

   • Decreased tuning to hand velocity in BCWH suggests different representation.

3. Neurons tuned to hand velocity in pole control code for robot velocity in brain-control. How do those neurons' modulation not lead to hand movements in brain-control? In other words how does the monkey inhibit hand movements during brain control? Suggests activity on the level of ensemble combination changes to accomodate it. The recorded population is small part, may be preferentially assigned weights to control the actuator, whereas the others operate normally. This is the topic of Ganguly and Carmena, 2011.

4. Neuronal ensemble represents robot velocities better during brain control supports optimal feedback control theory -- motor system as a stochastic feedback controller that optimizes only motor parameters necessary to achieve task goals (meh).

5. Once the neuronal ensemble starts to control a BMI, imperfections in open-loop model may be compensated by neuronal adaptations to improve control, concurs with (Taylor 2002) using adaptive BMI. Justifies using fixed decoder, and closed-loop decoder calibration stage (CLDA, ReFIT, Schwartz algs).

Further Questions

6. How exactly does the cortical plasticity process help? Very likely features of adaptations depend on requirements of BMI operations. If the task requires both limb movements and BMI operations, what will be the optimal neural system state? It's possible certain adapation states is only temporary to maintain performance.

   • PDs of many neurons become more similar in BCWOH -- may be maladaptive since PD diversity may be needed to improve directional encoding.

   • Shuffling procedures show increased correlations accompanies increased individual neuronal tuning...seems to contradict the previous point.

   • Therefore, neuronal ensemble controlling hte BMI may have been optimizing its performance by trading magnitude of tuning and the diversity of PD, or that increased correlation is only a temporary effect and will decrease with prolonged training (attention in learning stage increases synchrony).

Probably led to Orsborn & Carmena 2014's task of pressing a button while doing BMI center-out.

R-Squared, r, and Simple Linear Regression

Linear regression is perhaps the simplest and most common statistical model (very powerful nevertheless), and fits the model of the form:

This model makes some assumptions about the predictor variables, response variables, and their relationship:

1. Weak exogeneity - predictor variables are assumed to be error-free.

2. Linearity - response variables is a lienar combination of the predictor variables.

3. Constant variance - response variables have the same same variance in their errors, regardless of the values of the predictor variables. This is most likely invalid in many applications. Especially so when the data is instationary.

4. Independence of errors - assuming errors of response variables are uncorrelated with each other.

5. Lack of multicollinearity - in the predictors. Regularization can be used to combat this. On the other hand, when there is a lot of data, the collinearity problem is not as significant.

The most common linear regression is fitted with ordinary least-squares (OLS) method, where the sum of the squares of the differenceces between the observed responses in the dataset and those predicted by the model is minimized.

Two very common metrics associated with OLS linear regression is coefficient-of-determination $R^2$, and correlation coefficient, specifically Pearson correlation coefficient $r$.

The first measures how much variance is explained by the regression, and is calculated by $1-\frac{S{S}_{res}}{S{S}_{tot}}$. The second measures the the lienar dependence between the response and predictor variables. Both have value between 0 and 1. Say we fit linear regression and want to check how good that model is, we can get the predicted value $\hat{\mathbf{y}}=\mathbf{X}\beta$ and check the ${\mathbf{R}}^{\mathbf{2}}$ and $\mathbf{r}$ values between $\mathbf{y}$ and $\hat{\mathbf{y}}$. When a linear regression is fit using OLS, then the $R^2=r^2$.

However, if two vectors are linearly related but not fit using OLS regression, then they can have high $r$ but negative $R^2$. In MATLAB:

>> a = rand(100,1);
>> b = rand(100,1);
>> corr(a,b)

ans =

    0.0537

>> compute_R_square(a,b)

ans =

   -0.7875

>> c = a + rand(100,1);
>> corr(a,c)

ans =

    0.7010

>> compute_R_square(a,c)

ans =

   -2.8239

Even though a and c above are clearly linearly related, their $R^2$ value is negative. This is because the $1-\frac{S{S}_{res}}{S{S}_{tot}}$ definition assumes c is generated from an OLS regression.

According to Wikipedia,

Important cases where the computational definition of R2 can yield negative values, depending on the definition used, arise where the predictions that are being compared to the corresponding outcomes have not been derived from a model-fitting procedure using those data, and where linear regression is conducted without including an intercept. Additionally, negative values of R2 may occur when fitting non-linear functions to data.[5] In cases where negative values arise, the mean of the data provides a better fit to the outcomes than do the fitted function values, according to this particular criterion

This point really tripped me up when I was checking my decoder's performance. Briefly, I fit a Wiener filter to training data, apply on testing data, and then obtain the $R^2$ and $r$ between the predicted and actual response of the testing data. I expected $R^2=r^2$. However, $R^2$ was negative and its magnitude was not related to $r$ at all -- precisely for this reason.

In the case of evaluating decoder performance then, we can instead:

1. Fit a regression between $\mathbf{y}$ and ${\mathbf{y}}_{\mathbf{pred}}$, then check the $R^2$ of that regression. Note that when there is a lot of data, regardless of how weak the correlation is between them, the fit will be statistically significant. So while the value of $R^2$ is useful, its pvalue is not as useful.

2. Correlation coefficient $r$. However it is susceptible to outliers when there is not much data. This has been my go-to metric for evaluating neural tuning (as accepted in the field) and decoder performance.

3. Signal-to-Noise (SNR) ratio. As Li et al., 2009 pointed out, $r$ is scale and translational invariant, this means a vector $[1,2,3,4,5]$ and $[2,3,4,5,6]$ will have $r=1$, which does not capture this offset.

SNR is calculated as $\frac{var}{mse}$ (often converted to dB) where $var$ is the sample variance $y$, and $mse$ is the mean squared error of $y_{pred}$. This then measures how $y_{pred}$ 'tracks' $y$, as desired.

Classifier performance in primary somatosensory cortex towards implementation of a reinforcement learning based brain machine interface, submitted to Southern Biomedical Engineering Conference, 2016.

In the theme of RL-based BMI decoder, this paper evaluates classifiers for identifying the reinforcing signal (or ERS according to Millan). Evaluates the ability of several common classifiers to detect impending reward delivery within S1 cortex during a grip force match to sample task performed by monkeys.

Methods

1. Monkey trained to grip right hand to hold and match a visually displayed target.

2. PETHs from S1 generated, from 1) aligning with timing of visual cue denoting the impending result of trial (reward delivered or withheld), 2) aligning with timing of trial outcome (reward delievery or withold).

   PETH time extended 500ms after stimulus of interest using a 100ms bin width. (So only 500ms total signal, 5-bin vector samples).

3. Dimension-reduction of PETH via PCA. Use only the first 2 PCs, feed into classifiers including:

   • Naive Bayes
   • Nearest Neighbor (didn't specify how many k...or just one neighbor..)
   • Linear SVM
   • Adaboost
   • Quadratic Discriminant Analysis (QDA)
   • LDA

   Trained on 82 trials, tested on 55 trials. Goal is to determine if a trial is rewarding or non-rewarding.

Results

1. For all classifiers, post-cue accuracy higher than post-reward accuracy. Consistent with previous work (Marsh 2015) showing the presence of conditioned stimuli in reward prediction tasks shift rerward modulated activity in the brain to the earliest stimulus predictive of impending reward delivery.

2. NN performs the best out of all classifiers. But based on the sample size (55 trials) and the performance figures, not significantly better.

Takeaways

Results are good, but should have more controlled experiments to eliminate confounds...could be seeing an effect of attention level.

Also, still limited to discrete actions.

Neural correlates of decision variables in parietal cortex

Motivation

Sensory-motor reflex has been tacitly accepted by many older physiologists as an appropriate model for describing the neural processes that underlie complex behavior. More recent findings (in the 1990s) indicate that, at least in some cases, the neural events that connect sensation and movement may involve processes other than classical relfexive mechanisms and these data suppport theoretical approaches that have challenged reflex models directly.

Researchers outside physiology have long argued (since 1960s) for richer models of the sensory-motor process. These models invok3 the explicit representation of a class of decision variables, which carry information about the environment, are extracted in advance of response selection, aid in the interpretatin or processing of sensory data and are a prerquisite for rational decision making.

This paper describes a formal economic-mathematical approach for the physiological studey o fthe senosry-motor process/decision-making, in the lateral intra-parietal (LIP) area of the macaque brain.

Caveats

1. Expectation of reward magnitude result in neural modulations in the LIP. LIP is known to translate visual signals into eye-movement commands.

2. The same neurons are sensitive to expected gain and outcome probability of an action. Also correlates with subjective estimate of the gain associated with different actions.

3. These indicate neural basis of reward expectation.

Decision-Making Framework

The rational decision maker makes decision based on two environmental variables: the gain expected to result from an action, and the probability that the expected gain will be realized. The decision maker aims to maximize the expected reward.

Neurobiological models of the processes that connect sensation and action almost never propose the explicit representation of decision variables by the nervous system. Authors propose a framework containing

1. Current sensory data - reflect the observer's best estimate of the current state of the salient elements of the environment.

2. Stored representation of environmental contingencies - represent the chooser's assumptions about current environmental contingencies, detailing how an action affects the chooser.

This is a Bayesian framework.

Sensory-Motor integration in LIP

In a previous experiment, the authors concluded that visual attention can be treated as conceptually separable from other elements of the decision-making process, and that activity in LIP does not participate in sensory-attentional processing but is correlated with either outcome continguencies, gain functions, decision outputs or motor planning.

Experiments

1. Cue saccade trials: a change in the color of a centrally located fixation stimulus instructed subjects to make one of two possible eye-movement responses in order to receive a juice reward. Before the color change, the rewarded movement was ambiguous. The successive trials, the volume of juice delivered for each instructed response (expected gain), or the probability that each possible response would be instructed (outcome probability) are varied.

   Goal is to test whether any LIP spike rates correlate to variations in these decision variables.

2. Animals were rewarded for choosing either of two possible eye-movement responses. In subsequent trials, varied the gain that could be expected from each possible response. The frequency with which the animal chose each response is then an estimate of the subjective value of each option.

   Goal is to test whether any LIP activity variation correlates with the subjective estimate of decision variable.

Analysis

Experiment 1

For each condition: 1) keeping outcome probability fixed while varying expected gain, and 2) keeping expected gain fixed while varying outcome probability, a trial is divded into 6 200ms epochs. The activity during these epochs are then used to calculate regressions of the following variables:

1. Outcome probability.
2. Expected gain.
3. Type of movement made (up or down saccade).
4. Amplitude of saccade.
5. Average velocity of saccade.
6. Latency of saccade.

Multiple regression were done for each epoch, percentage of recorded neurons that show significant correlations between firing rate and the decision variables were calculated.

Average slope of regression against either decision variables were taken as a measure of correlation. The same regressio slopes were calculated for type of movement made as well.

For both expected gain and outcome probability, the regression slopes were significantly greater than 0, and greater than that for type of movement during early and late visual intervals, and decay during late-cue and movement. The opposite is seen for type of movement regression slopes.

Experiment 2

Similar analysis is done to evaluate neural correlate of subjective estimate of reward. Results show activity level correlates with the perceived gain associated with the target location within the neuron's receptive field. The time evolution of the regression slopes show a similar decay as before.

Consistent with Herrnstein's matching law for choice behavior, there was a linear relationship between the proportion of trials on which the animal chose the target inside the response field and the proportion of total juice available for gaze shifts to that target.

Animal's estimate of the relative value of the two choices is correlated with the activation of intraparietal neurons.

Discussions

1. It would seem obvious that there are neural correlates for decision variables, rather than the traditional sensory-motor reflex decision-making framework. But this may be solely in the context of investigating sensory areas..probably have to read Sherrington to know how he reached that conclusion.

2. The data in this paper can be hard to reconcile with traditional sensory-attentional models, which would attribute modulations in the activity of visually responsive neurons to changes in the behavior relevance of visual stimuli, which would then argue that the author's manipulations of expected gain and outcome probably merely altered the visual activity of intraparietal neurons. Explainations include:

   • Previous work show intraparietal neurons are insensitive to changes in the behavioral relevance of either presented stimulus when it is not the target of a saccade.

   • Experiment 2 showed that intraparietal neuron activity was modulated by changes in the estimated value of each response during the fixation intervals BEFORE the onset of response targets.

3. Modulations by decision variables early in trial. Late in trial more strongly modulated by movement plan. Suggests intraparietal cortex lies close to the motor output stages of the sensory-deicsion-motor process. Prefrontal cortex perhaps has reprsentation of decision variables throughout the trial.

Parietal Lobe: From Action Organization to Intention Understanding

Inferior parietal lobule (IPL) neurons were studied when monkeys performed motor acts embedded in different actions, and when they observed similar acts done by an experimenter. Most IPL neurons coding a specific act (e.g. grasping) showed markedly different activiations depending on contexts (e.g. grasping for eating or for placing).

Many motor IPL neurons also discharged during the observation of acts done by others. The context-dependence also applied to when these actions were observed.

These neurons fired during the observation of an act, before the beginning of the subsequent acts specifying the action - thus these neurons not only code the observed motor acts but also allow the observer to understand the agent's intentions.

Posterial parietal cortex has traditionally been considered as the association cortex. Besides "putting together" different sensory modalities, also

1. Codes motor actions
2. Provides the representations of these motor actions with specific sensory information.

Experiment

Set 1

Investigates neuron selectivity during actioin performance:

1. Monkey starts from a fixed position, reache for and grasp a piece of food located in front of it and brigh the food to mouth.
2. Monkey reach for and grasp an object, and place it in a box on the table.
3. Monkey reach for and grasp an object, and place it into a container near its mouth.

Most observed neurons discharged differentially with respect to context.

Control for confounds:

1. Grasp an identical piece of food in both conditions - rewarded with the same piece of food in situation 2.
2. Situation 3 aims to determine if arm kinematics determined the different firing behavior - turns out to be pretty much identical to situation 2.
3. Force exerted grasping - mechanical device that could hold the objects with two different strengths used, no change in neuron selectivity.

Set 2

Study 41 mirror neurons that discharge both during grapsing observation and grasping execution, in an experiment where the monkey observed an experimenter performing those actions.

Results:

1. Some neurons are selective for grasping to eat, others for grasping to place - most have the same specificity as when the monkey performed those actions.
2. Most neurons have the same selectivity regardless of whether the grasped object is food or solid. Some discharged more strongly in the placing conditions when the grasped object was food.

The authors suggest that while counter-intuitive, this should makes sense since motor acts are not related to one another independent of the global aim of the action but appear to form prewired intentional chains in which each motor act is faciliatted by the previously executed one. This reminds me of Churchland and Shenoy's theory about action planning as a trajectory in the Null-Space.

Consequences

The authors suggest that the monkey's decision on what to do with the object is undoubtedly made before the onset of the grasping movement. The action intentionis set before the beinning of the movements and is already reflected in the first motor act. This motor reflectioin of action intention and the chained motor organization of IPL neurons have profound consequences on [...] understanding the intention of others.

While I can see where he reaches this conclusion, and it would nicely explain why passive observation training for BMI works (even though mostly M1 and S1 neurons are recorded), but this explanation would make more sense if most of the difference in discharge rate happens BEFORE the monkey physically grasp the object/food...

It has been generally accepted that mirror neurons allow the observing individual to understand the goal of the observed motor act. The authors' interpretation then deduces that because the monkey knows the outcome of the motor act it executes, it recognizes the goal of the motor act done by another individual when this act triggers the same set of neurons that are active during the execution of that act.

Applications to BMI

This paper suggests mirror neurons as a mechanism for evaluation of intention, it also suggests that depending on context and object, a simple grasp action may be coded differently. Thus a BMI arm trained on passive observation of grasping an object to place might not work well when the user tries to grasp a food to eat. This is similar to when the Schwarz 10-DOF arm encountered, when the human subject had trouble grasping object. The problem only went away when the training was done in a virtual space.

Errare machinale est: the use of error-related potentials in braine-machine interfaces

This review focuses on electrophysiological correlates of error recognition in human brain, in the context of human EEG. Millan calls this error-related potentials, ErrPs. It has been proposed to use these signals to improve BMI, in the form of training BMI decoder via reinforcement learning.

Error-related brain activity

Early reports in EEG date back to early 1990's (Falkenstein et al., 1991; Gehring et al., 1993). Showed a characteristic EEG event-related potential elicited after subjects commited errors in a speed repsonse choice task, characterized by a negative potential deflection, termed the error-related negativity, appearing over fronto-central scalp areas at about 50-100ms after a subject's erroneous response. The negative copmonent is followed by a centro-parietal positive deflection.

Modulations of this latter component have been linked to the subject's awareness of the error. ERN amplitude seems to be modulated by the importance of erros in the given task. These signals have been shown to be quite reliable over time and across diferent tasks.

Very encouraging for reinforcement-learning based schemes.

fMRI, EEG-based source localization, and intra-cranial recordings suggest that the fronto-central ERP modulations commonly involve the medial-frontal cortex, specifically the anterior cingulate cortex (ACC). This is consistent with ACC being involved in reward-anticipation and decision-making.

Error-related potentials for BMI

Ferrez and Millan 2008: 2-class motor-imagery bsaed BMI controlling a cursor moving in discrete steps -- showed ERPs elicited after each command could be decoded as corresponding to the error or correct condition with an accuracy of about 80%. Only 5 subjects, but encouraging.

The ErrP classifiers maintained the same performance when tested several months after calibration - relatively stable signal form. Seemed to be mainly related to a general error-monitoring process, not task-dependent.

Possible Confounds (in EEG studies):

1. Eye movement contamination: Need to make sure location or movement of target stimuli are balanced.
2. Observed potentials are more related to the rarrity of the erroneous events (?)
3. Attential level: subjects tend to have smaller ErrP amplitude when simply monitoring the device than when they are controling it -- requires efficient calibration methods.

Error-Driven Learning

Trials in which the decoder performed correctly, as classified by ErrP can be incorporated into a learning set that can be then used to perform online retraining of the motor imagery (MI) classifier. Rather discrete, but good architecture.

Binary ErrP decoding can be affected substantially by false positive rate (i.e. correct BMI actions misclassified as errors). Solution is to use methods relying on probabilistic error signals, taking into account of the reliability of the ErrP decoder.

Millan has used ErrP to improve his shared-control BMI systems.

A learning-based approach to artificial sensory feedback leads to optimal integration

Motivation

Maximally precise movements are achieved by combining estimates of limb or target positiion from multiple sensory modalities, weighting each by its relative reliability. Current BMI relies on visual feedback alone, and would be hard to achieve fluid and precise natural movements. Artificial proprioception may improve performance of neuroprosthetics.

Goal

Artificial proprioception should be able to:

1. Provide sufficient information to alow competent performance in the absence of other sensory inputs.
2. Permit multisensory integration with vision to reduce movement variability when both signnals are available.
   Implemented via multichannel intracortical microstimulation. Different from most other works that take a biomimetic approach. It focuses not on reproducing natural patterns of activity but instead on taking advantage of the natural mechanisms of sensorimotor learning and plasiticity.

Hypothesize spatiotemporal correlations between a visual signal and novel artificial signal in a behavioral context would be sufficient for a monkey to learn to integrate the new modality.

Approach

Monkey looks at a screen, trying to move to an invisible target. The movement vector is the vector between the monkey's current hand position and the invisible target.

1. Visual feedback: The movement vector is translated to random moving-dot flow field. The coherence of the dot-field is the percentage of dots moving in the same direction. So 100% coherence dot-field would have the dots moving in the same direction as the movement vector.
2. ICMS: 96-channels array is implanted. 8 electrodes were stimulated on. During feedback, the movement vector is decomposed into 8 components in direction equally spaced around a circle, each corresponding to a stimulating electrode. Higher relative frequency on an electrode means greater component of the movement vector in that direction. Common frequency scaling factor corresponds to movement vector distance.

Experiments inluded:

1. Learning the task with only visual feedback, with different coherence to demonstrate performance differences due to changing sensory cue precision.
2. Learning the task with only ICMS feedback. Trained by providing ICMS during visual task to teach monkey to associate the spatial relationship. No changing coherence.
3. Performing the task with both.

Metrics:

1. Percent trials correct
2. Number of movement segments - small is better. Movement segment found by assuming sub-movements have bell-hsaped velocity profiles -- identify submovements by threshold crossings of the radio velocity plot of a trajectory.
3. Normalized movement time: by the distance to target.
4. Noramlized path length: Normalized the integrated path length by the distance from start to target.
5. Mean and Variance of the initial angle: Used to compare initial movement direction and movement vector.
6. Distance estimation: Assess the monkey's ability to estimate target distance from ICMS by regressing initial movement distance against target distance for ICMS-only trials, excluding trials for which the first movement segment was not the longest.

Evaluating Sensory Integration

Minimum variance integration, which is supposed to the optimal integration, predicts that in VIS+ICMS condition, as the visual coherence increases from 0% to 100%, the animals should transition from relying primarily on the ICMS cue to primarily on the visual cue -- under the model of minimum variance integration, each sensory cue should be wieghted inversely proportional to its variance.

Results

1. Monkey learns all three tasks.
2. VIS+ICMS outperforms VIS only for visual coherence less than 50%.
3. Optimal (minimum variance integration) integration of the two sensory modalities observed.

And only 8 electrodes!!