Measuring & Induing Neural Ativity Using Extraellular Fields I: Inverse systems approah Keith Dillon Department of Eletrial and Computer Engineering University of California San Diego 9500 Gilman Dr. La Jolla, CA 92093 kdillon@usd.edu Abstrat This projet onsiders the problem of trying to seletively sense and indue ativity in individual neurons within a network in a living organism, using a small array of eletrodes plaed some distane away. We onsider a variety of approahes to relate neural behavior to the potential or urrent at the eletrode array. Then we perform a simulation using the simple linear model with uniform ondutivity, and demonstrate the ability to resolve axons in the transverse diretion. We also demonstrate the use of this same approah to estimate a urrent at the eletrode array to ahieve, as best possible, a desired potential differene within the neural tissue. 1 Introdution The tetrode and related devies suh as the multitrode or multieletrode [1], are ommonly used for extraellular reording of neural ativity. Typially researhers make use of so-alled spike sorting algorithms to differentiate neurons by orrelating their ativity [2]. This projet may be onsidered as an effort to model the interation of suh devies with neurons using an inversesystems approah, effetively produing an image of neural ativity in spae. The neuron may be treated as a soure of urrents, whih are what we want to loate. The physis for this problem boils down to a problem in eletrostatis, whih we will onsider in the next setion. So we fous on problems that involve relating the urrent to the potential in a region, where knowledge of either the potential or the urrent may be limited to measurements at the boundary. We address ases both with onstant and varying ondutivity. In the simplest ase, of uniform ondutivity, we have a linear system of equations whih we go on to analyze via simulation in the subsequent setion. We also demonstrate how the system hanges as the assumptions are hanged or made more realisti. 2 Theory We assume extraellular fields fall within the quasistati regime, where oupling via magneti fields is insignifiant. Further, we assume the neural tissue to be Ohmi, i.e. a pure ondutor desribed by a real ondutivity tensor. Under these assumptions, the typial starting point [3] is Eq. (1.
( σ ( ( ρ ( r r = r (1 Where σ is the ondutivity, ϕ is the potential, and ρ is the harge density. We break the ondutivity out into onstant and varying parts, Then we have ( = + ( σ r σ δσ r (2 ( ( (( ( ( σ ( r ( δσ ( r ( r 2 ( r ( ( r ( r ρ r = σ + δσ r r = + = σ + δσ (3 Whih we write as a Poisson equation with foring funtion, ( ( r σ ( r σ ( r ( r = f ( r 2 1 1 = ρ δσ (4 The Green s funtion, whih we define as the solution to, ( δ ( 2 G r = 3 r (5 is well-known for equations of this form. With boundary onditions of zero field at infinity, we get, via Green s theorem, 1 (6 G ( r = 4π r Then using the Green s funtion, we an solve Eq. (4 as ( r = G ( r r f ( r dr 1 = σ ( { ( ( ( ( } G r r ρ r + δσ r r dr 1 1 = σ ( ρ ( ( ( ( σ δσ ( G r r r dr G r r r r dr (7 Where we have used the fat that the Green s funtion is shift-invariant. The first term on the r. right-hand side gives the potential in a onstant ondutivity, whih we all ( 0 From an inverse-sattering perspetive, is the inident field and the seond term is the 0 sattered field. The inident field we an ompute easily given the onstant part of the ondutivity. The sattered field term, however, depends on the total potential ( r, whih is unknown. Using the definition of the urrent-soure density, We write 0 as i ( ρ ( r r (8
1 ( = σ G ( ( ρ i( r = dr r r r r dr 0 4πσ r r (9 The most ommon approah to solving systems like this is to approximate the potential as resulting from a onstant ondutivity, i.e., ( r = 0 ( r + s ( r 0 ( r i ( r dr 4πσ r r (10 This is a linear system whih may either be solved for the potential given known urrent soures and ondutivity, or inverted to get the urrent soures given a known potential and known ondutivity. We an onsider the integral over the Greens funtion as a linear operator (a transfer funtion here, i ( r ( r dr = H i ( r πσ r r (11 4 H operates on a funtion over r (the region to be imaged and returns a funtion over r (the measured points. If we hoose basis funtions for these regions (e.g. delta funtions spaed on a grid we may form a linear system as v = Hi (12 This is the system we will onsider in this report. If we wish to estimate the urrent density, but the potential is only partly known, for example only at ertain points measured by leads, then we have an ill-posed inverse problem. Next we will derive some better approximations. To improve on the uniform-ondutivity approximation we go bak to Eq. (7, and now we write it as ( ( r δσ ( r ( r d i ( r = r dr 4πσ r r 4πσ r r (13 For the seond term we form the operator K whih we assume is known to get the system, σ ( r = i ( r + ( r H K (14 And we assume the domain and range of the operators is the same (i.e. the operators map the region to itself. If we have known index variations and know the potential everywhere, we an estimate the urrent via inversion of this equation, forming a linear system as before, ( σ σ I K v = Hi (15 If the ondutivity variations are unknown, but the potential is known everywhere, we an view the seond term in Eq. (13 as a linear operator that depends on (e.g. K ating on δσ, and form a linear system ( r = i ( r + δσ ( r H K (16
In terms of basis oeffiients, the system beomes v = Hi + K s H i = K s (17 Where s is the vetor representing the oeffiients of δσ on the basis. If the potential is only known on a limited number of points, and the potential over the region in the integral in (13 is assumed to be ompletely unknown, but the ondutivity is known, we have v = Hi + K v (18 d σ s Where tissue. v d is the (known potential at the detetor, and v is the (unknown potential in the neural t If the ondutivity is also unknown we have a quadrati problem. In operator form it would be ( r = i ( r + ( r ( r H K δσ (19 Where K is a new operator whih performs the onvolution with the Green s funtion and the r. Note that a gradient-based solver for divergene, applied to the produt of δσ ( r and ( this system would, via the produt rule for derivatives, solve the above linear systems. We may form an approximation analogous to the Born approximation of inverse sattering, by assuming the unknown potential values are equal to the potential in onstant medium, namely 0 ( r. Then we have the following variation on Eq. (17, where we use K in plae of 0 K. H i v = K s 0 (20 Now we demonstrate simulated results using the system of Eq. (11. 3 Results We model neurons in tissue as lines of urrent soures. By onsidering a typial ation potential in a simplified neuron we note that at any given point in time, the signal resulting from a single ation potential strethes aross hundreds of mirons if not millimeters, as depited in Fig. 1.
Figure 1: Axon urrent versus time simulated using NEURON [4]. We also find typial urrent levels to be in the tens to hundreds of miroamps. Our model of a neuron therefore, will be a line of urrent soures, whih we will assume to be onstant over the measurement regime. First we address the problem of measuring neural ativity via extraellular fields. We onsider the following geometry, depited in Fig. 2. The detetor is a two-dimensional plate at the y-z plane, with dimensions of 100 by 100 mirons. The region to be analyzed is the three dimensional volume adjaent to the plate in the positive x-diretion, extending 20 mirons deep. (a (b Figure 2: Simulation geometry. (a Detetor points where potential is measured, and (b variable points where urrent soure density is to be estimated. We use the simplified model of Eq. (11, whih in terms of the samples values is v = Hi (21 v is a vetor ontaining the potential at the points in Fig.?? (a, and i is a vetor ontaining the urrent density at the points in Fig. 2 (b. H is a matrix whih approximates the linear operation in Eq. (11. In the ase depited here, v is length 400 (the detetor grid is 20x20, while i is length
7600 (the variable grid is 19x20x20. Therefore H is 400x7600 and we have a signifiantly underdetermined problem. We an estimate a regularized solution using trunation of the singular-value deomposition (SVD [6]. If the SVD of H is T H = USV (22 Then we an form T i VSU % v (23 Where S % is a retangular diagonal matrix with the inverses of the nonzero singular values of H on the diagonal. The singular values of H are plotted in Fig. 3 in dereasing order. 10 0 10-1 10-2 0 50 100 150 200 250 300 350 400 Figure 3: Magnitude of singular values, normalized by largest. We note that they are all within two orders of magnitude of the largest, meaning that roughly a signal-to-noise ratio of 100 is neessary to ollet all available degrees of freedom of the measured data. We assume that we are able to do so in this report. Next we onsider the energy in the olleted signal versus depth. In Fig. 4 we have plotted the sum of the energy in the eigenvetors orresponding to the nonzero eigenvalues, integrated over the y-z diretion, versus the depth diretion.
10 0 10-1 10-2 10-3 2 4 6 8 10 12 14 16 18 x (µ m Figure 4: Relative energy in singular values versus depth. This result demonstrates that most of the signal that results from deeper soures is not transferred to the detetor at deeper distanes, i.e. it is projeted of the nullspae of H. Now we will perform a simulation using Eq. (23 to estimate the urrent soure density. We use the onfiguration of Fig. 5 (a as our unknown soure density, onsisting of 3 vertial lines at 7 miron depth, and 3 horizontal lines at 14 miron depth. At the detetor we see the simulated result of Fig. 5 (b. (a (b 2 4 6 8 10 12 14 16 18 20 5 10 15 20 200 190 180 170 160 150 140 130 120 110 100 Figure 5: (a simulated urrent density, (b voltage measured at the grid of detetor points (in volts. We solve the inverse system using the SVD approah, and get the estimate of Fig. 5.
Figure 6: Inversion result, given in ross setions parallel to detetor at inreasing depth. We find that we are able to resolve the soures in the transverse diretions (y-z, but that depth information is basially gone. Assuming that the reonstrution always makes the mistake of attributing the soure to the plane nearest the detetor, we now onsider the image only at that plane, as we vary the soure depth. In Fig. 7, we show the potential measured at the detetor as we vary the depth of a vertial line soure.
Figure 7: Potential measured at detetor for vertial line soures at inreasing depth. In Fig. 8 we find that we are able to reliably resolve the soure in the y-z diretion by simply looking at the first plane of the inverse estimate.
Figure 8: Estimate of Current soure density at plane adjaent to detetor, for soures at inreasing depth. Next we onsider the problem of trying to indue urrents rather than measure them. The problem is essentially the same as above, but now the plate detetor is the unknown urrent soure, and the imaged region is the desired potential. Our goal is to reate a small potential differene. We presume a rapid hange in the depth diretion would be pratially impossible given the system s invariane with depth, so as our target we use a pair of points of opposite voltage adjaent in the z- diretion. In Fig. 9, we show the urrent soure density needed at the soure plate to ahieve the best possible approximation to a small dipole at different distanes.
(a (b 2 x 10-6 1.5 2 x 10-9 3 4 1 4 2 6 6 8 0.5 8 1 10 0 10 0 12 12 14-0.5 14-1 16 18-1 16 18-2 20-1.5 5 10 15 20 20 5 10 15 20-3 Figure 9: Needed urrent soure densities to ahieve desired potential. (a at 2 miron depth, (b at 18 miron depth. To find what we atually expet to ahieve, we solve the forward system of Eq. (21 with the above urrent soures, and get the results in Figs. 10 and 11. Figure 10: Potential resulting from urrent soure of Fig. 8 (a at soure plate, displayed in rosssetions at inreasing depth.
Figure 11: Potential resulting from urrent soure of Fig. 8 (b at soure plate, displayed in rosssetions at inreasing depth. As we should have expeted, we find we have minimal ability to ontrol variation in depth. 4 Disussion We found promising results in our ability to resolve urrent soures in the transverse diretion, with basially zero resolution in the depth diretion. Effetively we have a projetion of the urrent soure density in the depth diretion. This suggests a form of tomography, whereby we might perform this imaging proess over a range of angles and ombine the results to reonstrut in three dimensions. The design problem of produing as lose to possible a desired potential appears more hallenging. Our preliminary results suggest we may be able to provide a better resolution of the effet in the transverse diretions, over some depth. But the effet in a more realisti senario may be lost. Further, our real interest is in induing urrent into a neuron whih means we must onsider the lower ondutivity of the membrane versus the extraellular medium. Both these issues an perhaps be addressed by using the more sophistiated model of Eq. (20.
Referenes [1] Blanhe, T.J., Spaek, M.A., Hetke, J.F., & Swindale, N.V. (2005 Polytrodes: High-Density Silion Eletrode Arrays for Large-Sale Multiunit Reording. J Neurophysiol 93: 2987 3000. [2] Lewiki, M.S. (1998 A review of methods for spike sorting: the detetion and lassifiation of neural ation potentials. Network: Computation in Neural Systems. 9:R53-R78. [3] Nunez, P.L., & Srinivasan, R. (2006 Eletri fields of the brain: the neurophysis of EEG, 2e. New York: Oxford. [4] Carnevale, N.T. and Hines, M.L. (2006 The NEURON Book. Cambridge, UK: Cambridge University Press. [5] Hansen, P.C.. (1987 The trunated SVD as a method for regularization. BIT 27: 534-553.