In vivo ephaptic coupling allows memory network formation

Abstract It is increasingly clear that memories are distributed across multiple brain areas. Such “engram complexes” are important features of memory formation and consolidation. Here, we test the hypothesis that engram complexes are formed in part by bioelectric fields that sculpt and guide the neural activity and tie together the areas that participate in engram complexes. Like the conductor of an orchestra, the fields influence each musician or neuron and orchestrate the output, the symphony. Our results use the theory of synergetics, machine learning, and data from a spatial delayed saccade task and provide evidence for in vivo ephaptic coupling in memory representations.

where we linearized ( , ) ( , ) . The connectivity components wi are found below.
The principal axes Hi of the neural field (1) are given by the spatial derivatives of the neural activity (cf.last line in Equation (1) in main text).These contain temporal information, that is, fluctuations around baseline activity at different spatial scales, see (Pinotsis et al., 2017)  We then assumed that connectivity components were sampled from a normal distribution (0, ) Q w N I .This ensured that they were uncorrelated.Considered mean-centered activity, we obtain Equation (2).We then used a Restricted Maximum-Likelihood (ReML) algorithm to obtain Bayesian optimal values for the connectivity components k w (Harville, 1977b).This algorithm optimizes the objective function known as Free Energy, F, given by Equation (3).
This expression of the Free Energy obtains from the sum log ( ) ( ) (0, ) under a Laplace assumption, where Q is the approximate posterior and after substituting 21 ( , ) , see also (Friston, 2008).

II. Bidomain model of the ensemble electric field
According to the theory of electromagnetism, the discontinuity between extracellular and intracellular potential (see Figure 1C) gives rise to dipole sources with moments (Jackson, Here a p is the moment of a neural ensemble whose center is at location ( , )   aa xy , r is the brain resistivity with 2.2 Ohm r = (Rush and Driscoll, 1969) and we have assumed that the number of neurons is large and that each cell is very small compared to the distance at which the LFP electrode is placed.Also, the current density ( , ) Equation (II.4) is the same as Equation ( 5) in main text where the function () Wk is defined in the second line.12)

III. Constituent matrices of matrices M and D appearing in Equation (
IV. Derivation of Equation ( 6) Below, starting from Equation ( 5), we derive Equation ( 6).We first assumed that the LFP electrode is at a large distance compared to the size of the neural ensemble.In other words, the radius a of the intracellular fiber (grey) is very small compared to the vertical distance to the location of the LFP electrode, , 0. a y a   This is shown by a squashed grey cylinder in Figure 1D.Then, the distance from the surface of the intracellular fiber is approximately equal to R (Clark and Plonsey, 1968;Plonsey, 1974).Applying the convolution theorem after noting that the inverse Fourier transform (FT -1 ) of 0 ( ) K k y is ½(y 2 +x 2 ), Equation ( 4) can be written as An alternative way to derive this expression is to start from the general expression of EP in electromagnetism for sources distributed across the intracellular fiber (Jackson, 1999) e V x y R g x y dxdy and consider the density ( , ) ( ) , that describes charges distributed across an electrical fiber as a result of the transmembrane current 1/ m i rV  . Also, dxdy is the elementary source volume element.To calculate the integral in Equation (IV.1), we expand it in multipoles following standard theory of electromagnetism (Jackson, 1999).
Multipole expansions use Legendre polynomials Pm= Pm(r,ψ) that are often expressed in spherical coordinates (r,ψ), see (Figure 1D).Equation (IV.1) can be rewritten as .Without loss of generality, we consider a measurement point vertical to the ensemble and at fixed distance r>0, for which ψ=0 (Figure 1C).This simplifies the numerical expressions in Equation (IV.6), which now reads for details.The name principal axes derives from training the model (2) with a PCA-like algorithm using the cost function (3).For the ephaptic model discussed in main text, the principal axes are the index i correspond to derivatives of different orders.iH , i=1,2,3… are called the axes of order i, that is, first, second, third axis etc. Principal axes are matrixvalued functions of dimensionality T N xT , where T N is the number of trials and T is the length of the raw LFP time series.We call each entry in these matrices the axis strength.This corresponds to an instantaneous scale factor with which the corresponding component strength must be multiplied to reconstruct the observed LFP.Across-trial averages of principal axes for FEF and SEF were shown in Figures 2 and 3 of (Pinotsis et al., 2017b).

m
where Ω is the total volume of the ensemble.Neglecting ephaptic interactions me VV , and the extracellular electric potential generated by the current density ( , ) is the conductivity of the extracellular space, and R is the distance between the current source at the point '( , ) a P x a of the neural ensemble and the point( , )   xy in the extracellular space where we measure e V , i.e. the location of the LFP electrode, 1C.Here, a is the radius of the grey cylinder in the figure.According to the bidomain model, Equation (II.3) can be written as (Henriquez, 1993; Roth, Vk is the Fourier Transform of the transmembrane potential m V and FT -1 is its 1 is assumed to occupy a fiber of length L. The density g(x) is zero outside the patch, that is, for /2 xL  and the integral in the above expression is defined of the interval [-L/2,L/2].Then, odd terms in the above series vanish.Legendre polynomials of even order are given by the following expressions (Abramowitz et al.evaluate the integrals of the first few terms in the series appearing in Equation (IV.4).This yields the following multipole expansion for the term corresponds to a monopole and we have neglected terms of order five and above.Equation (IV.6) provides an algebraic expression connecting the resting state EP value e S V during memory delay to the second derivative of the transmembrane potential m V trial the remembered stimulus changes.Thus the EP and the corresponding EF also change, see(Pinotsis and Miller, 2022b)  for details.Because Equation (IV.7) provides the resting value, it needs to be extended into an evolution equation that describes the relaxation process of the EP.We thus assumed a simple fixed point equation 1