Definitions of state variables and state space for brain-computer interface

Abstract

Neocortical state variables are defined and evaluated at three levels: microscopic using multiple spike activity (MSA), mesoscopic using local field potentials (LFP) and electrocorticograms (ECoG), and macroscopic using electroencephalograms (EEG) and brain imaging. Transactions between levels occur in all areas of cortex, upwardly by integration (abstraction, generalization) and downwardly by differentiation (speciation). The levels are joined by circular causality: microscopic activity upwardly creates mesoscopic order parameters, which downwardly constrain the microscopic activity that creates them. Integration dominates in sensory cortices. Microscopic activity evoked by receptor input in sensation induces emergence of mesoscopic activity in perception, followed by integration of perceptual activity into macroscopic activity in concept formation. The reverse process dominates in motor cortices, where the macroscopic activity embodying the concepts supports predictions of future states as goals. These macroscopic states are conceived to order mesoscopic activity in patterns that constitute plans for actions to achieve the goals. These planning patterns are conceived to provide frames in which the microscopic activity evolves in trajectories that adapted to the immediate environmental conditions detected by new stimuli. This circular sequence forms the action-perception cycle. Its upward limb is understood through correlation of sensory cortical activity with behavior. Now brain-machine interfaces (BMI) offer a means to understand the downward sequence through correlation of behavior with motor cortical activity, beginning with macroscopic goal states and concluding with recording of microscopic MSA trajectories that operate neuroprostheses. Part 1 develops a hypothesis that describes qualitatively the neurodynamics that supports the action-perception cycle and derivative reflex arc. Part 2 describes episodic, “cinematographic–spatial pattern formation and predicts some properties of the macroscopic and mesoscopic frames by which the embedded trajectories of the microscopic activity of cortical sensorimotor neurons might be organized and controlled.

Appendix 1. Wave-pulse statistical relations

A continuous record is digitized at 1 KHz simultaneously of a spike train of a single neurons and the ECoG at the cortical site of penetration (Freeman 1975/2004). The ECoG is filtered in the pass band of the desired oscillatory frequency, e.g., 20–0 Hz. The spike train is accumulated to give n _p ~10,000 pulses expressed as a series of 0’s and 1’s. The N values of ECoG values are normalized to zero mean and unit SD; an amplitude histogram is divided into 61 bins centered at 0 and ranging between ± 3 SD in steps of 0.1 SD. At each time step the question is asked, is there a pulse in any bin between T = 0 and T = ±T _r time steps (e.g., T _r = ±25 ms preceding and following T = 0 ms). A 1-D table of the pulse occurrences at each amplitude is accumulated at T = 0, p(v). The n _v (v) values in each bin, n _v , are divided by the total number of pairs to get the probability density for amplitude at T = 0:

$$ P\left( V \right) = {{n_v \left( v \right)} \mathord{\left/ {\vphantom {{n_v \left( v \right)} N}} \right. \kern-\nulldelimiterspace} N} $$

((1))

The number of pulses in each bin, n _p (p,v) is divided by the total number of pairs to give the joint pulse-amplitude probability density at T = 0:

$$ P\left( {p \cap v} \right) = {{np\left( {p,v} \right)} \mathord{\left/ {\vphantom {{np\left( {p,v} \right)} N}} \right. \kern-\nulldelimiterspace} N} $$

((2))

The pulse probability density is divided by the amplitude probability density to give the pulse probability conditional on amplitude at T = 0:

$$ P\left( {p|v} \right) = {{P\left( {p \cap v} \right)} \mathord{\left/ {\vphantom {{P\left( {p \cap v} \right)} {p\left( v \right)}}} \right. \kern-\nulldelimiterspace} {p\left( v \right)}} $$

((3))

The algorithm is repeated at each time lag between − T _r and +T _r to get the pulse probability conditional on time and amplitude in 2-D, which is then normalized by dividing the function by the grand mean pulse probability, P _o :

$$ P\left( {p|T \cap v} \right) = P\left( {p \cap T \cap v} \right)p\left( v \right) $$

((4))

The function is normalized by dividing all values by the grand mean pulse probability, P _o to get the normalized conditional pulse probability (NCPD):

$$ P_n \left( {p|T \cap v} \right) = {{P\left( {p|T \cap v} \right)} \mathord{\left/ {\vphantom {{P\left( {p|T \cap v} \right)} {P_o }}} \right. \kern-\nulldelimiterspace} {P_o }} $$

((5))

The time dependence of the NCPD is found by averaging across the upper third of the range for v > 0, giving the pulse probability wave that is comparable to the autocorrelation of the filtered ECoG or LFP:

$$ P_{n,v} \left( T \right) = {1 \mathord{\left/ {\vphantom {1 k}} \right. \kern-\nulldelimiterspace} k}\sum {P_n \left( {p|T \cap v} \right)} , SD \leqslant v_k \geqslant 3SD $$

((6))

The sigmoid curve is the NCPD on amplitude is estimated by averaging over lag times at k values where the deviation of P _v (T) above zero is maximal:

$$ P_{n,t} \left( T \right) = {1 \mathord{\left/ {\vphantom {1 k}} \right. \kern-\nulldelimiterspace} k}\sum {P_n \left( {p|T \cap v} \right)} , P\left( {T_k } \right) > > P_o $$

((7))

The sigmoid curve is fitted to the data (Fig. 1A) in order to evaluate the upper asymptote, Q _m , as given in the equation inset with the data. The asymptote varies in proportion to the degree of arousal, and it has differing mean values for differing populations in the olfactory and limbic systems. The forward gain of the population is given by the derivative of the sigmoid curve, dq/dv. Two examples are shown for Q _m  = 2 in behavioral rest and Q _m  = 5 in arousal (Freeman 2000) for comparison with the numerical derivative in Fig. 1A. The maximal gain, v _max = ln Q _m from the second derivative set to 0, is displaced to the excitatory side. This asymmetry underlies the input-dependent nonlinearity of cortical dynamics, which is required for the destabilization in spontaneous breaking of symmetry by state transitions (Freeman and Vitiello 2006). From Freeman (1979; reprinted Ch. 10, 2000)