Brain Machine Interfaces: Modeling Strategies for Neural Signal Processing

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Brain Machine Interfaces: Modeling Strategies for Neural Signal Processing

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Brain Machine Interfaces: Modeling Strategies for Neural Signal Processing

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Brain Machine Interfaces: Modeling Strategies for Neural Signal Processing

Jose C. Principe, Ph.D.

Distinguished Professor ECE, BME

Computational NeuroEngineering Laboratory

Electrical and Computer Engineering Department

University of Florida

www.cnel.ufl.edu

principe@cnel.ufl.edu

- A man made device that either substitutes a sensory input to the brain, repairs functional communication between brain regions or translates intention of movement.

- Sensory (Input BMI): Providing sensory input to form percepts when natural systems are damaged.
- Ex: Visual, Auditory Prosthesis

- Motor (Output BMI): Converting motor intent to a command output (physical device, damaged limbs)
- Ex: Prosthetic Arm Control

- Cognitive BMI: Interpret internal neuronal state to deliever feedback to the neural population.
- Ex: Epilepsy, DBS Prosthesis
Computational Neuroscience and Technology developments are playing a larger role in the development of each of these areas.

- Ex: Epilepsy, DBS Prosthesis

General Architecture

BCI (BMI)bypasses the brain’s normal pathways of peripheral nerves (and muscles)

J.R. Wolpaw et al. 2002

BRAINMACHINE

INTENT

ACTION

Decoding

PERCEPT

STIMULUS

Coding

Neural InterfacePhysical Interface

Need to understand how brain processes information.

- Brain is an extremely complex system
- 1012 neurons
- 1015 synapses
- Specific interconnectivity

- The choice and availability of brain signals and recording methods can greatly influence the ultimate performance of the BMI.
The level of BMI performance may be attributed to selection of electrode technology, choice of model, and methods for extracting rate, frequency, or timing codes.

Coarse(mm)

http://ida.first.fhg.de/projects/bci/bbci_official/

Moran

Develop a experimental paradigm with a nested hierarchy for studying neural population dynamics.

Least Invasive

EEG

NRG IRB Approval for Human Studies

ECoG

NRG IACUC Approval for Animal Studies

Microelectrodes

Highest Resolution

- BMIs --- Invasive, work with intention of movement
- Spike trains, field potentials, ECoG
- Very specific, potentially better performance

- EEG
- Very small bandwidth

- Integration of probabilistic models of information processing with the neurophysiological reality of brain anatomy, physiology and purpose.
- Need to abstract the details of the “wetware” and ask what is the purpose of the function. Then quantify it in mathematical terms.
- Difficult but very promising. One issue is that biological evolution is a legacy system!
- BMI research is an example of a computational neuroscience approach.

- NeoCortical Brain Areas Related to Movement

Posterior Parietal (PP) – Visual to motor transformation

Premotor (PM) and Dorsal Premotor (PMD) -

Planning and guidance (visual inputs)

Primary Motor (M1) – Initiates muscle contraction

- Two different levels of neurophysiology realism
- Black Box models – no realism, function relation between input desired response
- Generative Models – minimal realism, state space models using neuroscience elements

Accessing 2 types of signals (cortical activity and behavior) leads us to a general class of I/O models.

Data for these models are rate codes obtained by binning spikes on 100 msec windows.

- Optimal FIR Filter – linear, feedforward
- TDNN – nonlinear, feedforward
- Multiple FIR filters – mixture of experts
- RMLP – nonlinear, dynamic

Consider a set of spike counts from M neurons, and a hand position vector dC (C is the output dimension, C = 2 or 3). The spike count of each neuron is embedded by an L-tap discrete time-delay line. Then, the input vector for a linear model at a given time instance n is composed as x(n) = [x1(n), x1(n-1) … x1(n-L+1), x2(n) … xM(n-L+1)]T, xLM, where xi(n-j) denotes the spike count of neuron i at a time instance n-j.

A linear model estimating hand position at time instance n from the embedded spike counts can be described as

where yc is the c-coordinate of the estimated hand position by the model, wji is a weight on the connection from xi(n-j) to yc, and bc is a bias for the c-coordinate.

x1(n)

yx(n)

…

…

…

z-1

z-1

z-1

z-1

yy(n)

xM(n)

yz(n)

In a matrix form, we can rewrite the previous equation as

where y is a C-dimensional output vector, and W is a weight matrix of dimension (LM+1)C. Each column of W consists of [w10c, w11c, w12c…, w1L-1c, w20c, w21c…, wM0c, …, wML-1c]T.

For the MIMO case, the weight matrix in the Wiener filter system is estimated by

R is the correlation matrix of neural spike inputs with the dimension of (LM)(LM),

where rij is the LL cross-correlation matrix between neurons i and j (i ≠ j), and rii is the LL autocorrelation matrix of neuron i.

P is the (LM)C cross-correlation matrix between the neuronal bin count and hand position, where pic is the cross-correlation vector between neuron i and the c-coordinate of hand position. The estimated weights WWiener are optimal based on the assumption that the error is drawn from white Gaussian distribution and the data are stationary.

The predictor WWiener minimizes the mean square error (MSE) cost function,

Each sub-block matrix rij can be further decomposed as

where rij() represents the correlation between neurons i and j with time lag . Assuming that the random process xi(k) is ergodic for all i, we can utilize the time average operator to estimate the correlation function. In this case, the estimate of correlation between two neurons, rij(m-k), can be obtained by

The cross-correlation vector pic can be decomposed and estimated in the same way, substituting xj by the desired signal cj.

From the equations, it can be seen that rij(m-k) is equal to rji(k-m). Since these two correlation estimates are positioned at the opposite side of the diagonal entries of R, the equality leads to a symmetric R.

The symmetric matrix R, then, can be inverted effectively by using the Cholesky factorization. This factorization reduces the computational complexity for the inverse of R from O(N3) using Gaussian elimination to O(N2) where N is the number of parameters.

- Normalized LMS with weight decay is a simple starting point.
- Four multiplies, one divide and two adds per weight update
- Ten tap embedding with 105 neurons
- For 1-D topology contains 1,050 parameters (3,150)
- Alternatively, the Wiener solution

- The first layer is a bank of linear filters followed by a nonlinearity.
- The number of delays to span I second
- y(n)= Σ wf(Σwx(n))
- Trained with backpropagation
- Topology contains a ten tap embedding and five hidden PEs– 5,255 weights (1-D)

Principe, UF

- Multiple adaptive filters that compete to win the modeling of a signal segment.
- Structure is trained all together with normalized LMS/weight decay
- Needs to be adapted for input-output modeling.
- We selected 10 FIR experts of order 10 (105 input channels)

d(n)

- Spatially recurrent dynamical systems
- Memory is created by feeding back the states of the hidden PEs.
- Feedback allows for continuous representations on multiple timescales.
- If unfolded into a TDNN it can be shown to be a universal mapper in Rn
- Trained with backpropagation through time

Data

Task 1

- 2 Owl monkeys – Belle, Carmen
- 2 Rhesus monkeys – Aurora, Ivy
- 54-192 sorted cells
- Cortices sampled: PP, M1, PMd, S1, SMA
- Neuronal activity rate and behavior is time synchronized and downsampled to 10Hz

Task 2

- Train the adaptive system with neuronal firing rates (100 msec) as the input and hand position as the desired signal.
- Training - 20,000 samples (~33 minutes of neuronal firing)
- Freeze weights and present novel neuronal data.
- Testing - 3,000 samples – (5 minutes of neuronal firing)

Signal to error ratio (dB)

Correlation Coefficient

(average)

(max)

(average)

(max)

LMS

0.8706

7.5097

0.6373

0.9528

Kalman

0.8987

8.8942

0.6137

0.9442

TDNN

1.1270

3.6090

0.4723

0.8525

Local Linear

1.4489

23.0830

0.7443

0.9748

RNN

1.6101

32.3934

0.6483

0.9852

- Based on 5 minutes of test data, computed over 4 sec windows (training on 30 minutes)

- When the fitting error is above chance, a sensitivity analysis can be performed by computing the Jacobian of the output vector with respect to each neuronal input i
- This calculation indicates which inputs (neurons) are most important for modulating the output/trajectory of the model.

Identify the neurons that affect the output the most.

Feedforward Linear Eq.

Feedforward RMLP Eqs.

General form of Linear Sensitivity

General form of RMLP Sensitivity

Decay trend appears in all animals and behavioral paradigms

Tuning

Sensitivity

Significance: Sensitivity analysis through trained models automatically delivers deeply tuned cells that span the space.

How does each cortical area contribute to the reconstruction of this movement?

Train 15 separate RMLPs with every combination of cortical input.

- Analysis is based on the time embedded model
- Correlation with desired is based on a linear filter output for each neuron

- Utilize a non-stationary tracking algorithm
- Parameters are updated by LMS

- Build a spatial filter
- Adaptive in real time
- Sparse structure based on regularization for enables selection

z-1

z-1

z-1

z-1

x1(n)

w11

y1(n)

//

c1

w1L

y2(n)

c2

…

…

xM(n)

cM

wM1

yM(n)

//

wML

Adapted by on-line LAR

(Kim et. al., MLSP, 2004)

Adapted by LMS

x1

k

. . .

y

r= y-X = y

Find argmaxi |xiTr|

i=0

xj

xj

xk

r= y-X = y-xjj

Adjustj s.t.

k, |xkTr|=|xiTr|

r= y-(xjj+ xkk)

Adjustj & k s.t.

q, |xqTr|=|xkTr|=|xiTr|

j

j

- Tap weights for every time lag is updated by LMS
- Then, the spatial filter coefficients are obtained by on-line version of least angle regression (LAR) (Efron et. al. 2004)

Hand Trajectory

(z)

Early

Part

Neuronal Channel Index

Late

Part

- Use partial information about the physiological system, normally in the form of states.
- They can be either applied to binned data or to spike trains directly.
- Here we will only cover the spike train implementations.
Difficulty of spike train Analysis:

Spike trains are point processes, i.e. all the information is contained in the timing of events, not in the amplitude fo the signals!

Build an adaptive signal processing framework for BMI decoding in the spike domain.

- Features of Spike domain analysis
- Binning window size is not a concern
- Preserve the randomness of the neuron behavior.
- Provide more understanding of neuron physiology (tuning) and interactions at the cell assembly level
- Infer kinematics online
- Deal with nonstationary
- More computation with millisecond time resolution

Time-series model

State

cont. observ.

Prediction

Updating

P(state|observation)

- State space representation
First equation (system model) defines a first order Markov process.

Second equation (observation model) defines the likelihood of the observations p(zt|xt) . The problem is completely defined by the prior distribution p(x0).

Although the posterior distribution p(x0:t|u1:t,z1:t) constitutes the complete solution, the filtering density p(xt|u1:t, z1:t) is normally used for on-line problems.

The general solution methodology is to integrate over the unknown variables (marginalization).

- There are two stages to update the filtering density:
- Prediction (Chapman Kolmogorov)
System model p(xt|xt-1) propagates into the future the posterior density

- Update
Uses Bayes rule to update the filtering density. The following equations are needed in the solution.

- Prediction (Chapman Kolmogorov)

For Gaussian noises and linear prediction and observation models, there

is an analytic solution called the Kalman Filter.

Continuous Observation

Linear

Neuron tuning function

Kinematic State

Firing rate

Linear

Gaussian

Prediction

P(state|observation)

Updating

[Wu et al. 2006]

In general the integrals need to be approximated by sums using Monte Carlo integration with a set of samples drawn from the posterior distribution of the model parameters.

Continuous Observation

Exponential

Neuron tuning function

Kinematic State

Firing rate

Linear

nonGaussian

Prediction

P(state|observation)

Updating

[Brockwell et al. 2004]

v

x

k-1

k-1

x

k

F

k

(

)

=

,

=

(

)

,

n

z

H

x

k

k

k

k

Tuning function

Neural Tuning function

Kinematics

state

Multi-spike trains observation

Decoding

Kinematic dynamic model

Key Idea: work with the probability of spike firing which is a

continuous random variable

Poisson Model

nonlinear

Point process

Neuron tuning function

Kinematic State

spike train

Linear

Gaussian

Prediction

P(state|observation)

Updating

[Brown et al. 2001]

nonlinear

Point process

Neuron tuning function

Kinematic State

spike train

nonLinear

nonGaussian

Prediction

P(state|observation)

Updating

Sequential Estimate PDF

[Wang et al. 2006]

- STEP 1. Preprocessing
1. Generate spike trains from stored spike times 10ms interval, (99.62% binary train)

2. Synchronize all the kinetics with the spike trains.

3. Assign the kinematic vector to reconstruct.

X=[position velocity acceleration]’

(more information, instantaneous state avoid error accumulation,

less computation)

kinematics

Neural spike trains

Encoding

(Tuning)

- A example of a tuned neuron
- Metric: Tuning depth:
- how differently does a neuron fire across directions?
- D=(max-min)/std (firing rate)

Neuron 72: Tuning Depth 1

kinematics direction angle

Information

neural spikes

velocity

spikes

Linear filter

nonlinear f

Poisson model

- Neural firing Model
- Assumption :
generation of the spikes depends only on the kinematic vector we choose.

- Spike Triggered Average (STA)
- Geometry interpretation

2nd Principal component

1st Principal component

Ref: Paradoxical cold

[Hensel et al. 1959]

- Consider the neuron as an inhomogenous Poisson point process
- Observing N(t) spikes in an interval DT, the posterior of the spike model is
- The probability of observing an event in Dt is
- And the one step prediction density (Chapman-Kolmogorov)
- The posterior of the state vector, given an observation DN

- Monte Carlo Methods are used to estimate the integral. Let represent a random measure on the posterior density, and represent the proposed density by
- The posterior density can then be approximated by
- Generating samples from using the principle of Importance sampling
- By MLE we can find the maximum or use direct estimation with kernels of mean and variance

Posterior density at a time index

lag

Step 3: Information Estimated Delays

Figure 3-14 Mutual information as function of time delay for 5 neurons.

For 185 neurons, average delay is 220.108 ms

Neural Tuning function

Kinematic State

spike trains

Prediction

Updating

P(state|observation)

NonGaussian

Results comparison

Table 3-2 Correlation Coefficients between the Desired Kinematics and the Reconstructions

Table 3-3 Correlation Coefficient Evaluated by the Sliding Window

[Sanchez, 2004]

- Our results and those from other laboratories show it is possible to extract intent of movement for trajectories from multielectrode array data.
- The current results are very promising, but the setups have limited difficulty, and the performance seems to have reached a ceiling at an uncomfortable CC < 0.9
- Recently, spike based methods are being developed in the hope of improving performance. But difficulties in these models are many.
- Experimental paradigms to move the field from the present level need to address issues of:
- Training (no desired response in paraplegic)
- How to cope with coarse sampling of the neural population
- How to include more neurophysiology knowledge in the design