Dynamic Moment Analysis of the Extracellular Electric Field of a Biologically Realistic Spiking Neuron
DDynamic Moment Analysis of theExtracellular Electric Field of a Biologically
Realistic Spiking Neuron
Joshua N. Milstein [email protected]
California Institute of TechnologyPasadena, CA 91125Christof Koch [email protected]
California Institute of TechnologyPasadena, CA 91125October 25, 2018
Abstract
Based upon the membrane currents generated by an action potential in a r X i v : . [ q - b i o . N C ] N ov biologically realistic model of a pyramidal, hippocampal cell withinrat CA1, we perform a moment expansion of the extracellular fieldpotential. We decompose the potential into both inverse and classicalmoments and show that this method is a rapid and efficient way tocalculate the extracellular field both near and far from the cell body.The action potential gives rise to a large quadrupole moment that con-tributes to the extracellular field up to distances of almost 1 cm. Thismethod will serve as a starting point in connecting the microscopicgeneration of electric fields at the level of neurons to macroscopic ob-servables such as the local field potential. Since the pioneering work of Hodgkin and Huxley in the early fifties (Hodgkin& Huxley, 1952d) on the initiation and propagation of action potentialswithin the squid giant axon, there has been significant progress in our un-derstanding of brain function at the level of the single neuron (Koch, 1999).Unfortunately, it has proved difficult to connect function at this microscopicscale to more global, large-scale brain function. In this paper, we work towardthis goal by developing a physiologically accurate model of the extracellularfield of a single neuron which may be efficiently employed to model the fieldassociated with very large numbers of neurons.The dominant means of rapid communication among neurons is throughchemically or electrically mediated synapses. Ephatic interactions, where2ommunication is directly via an electric field, may occur in nerves that havebeen crushed or damaged by neurodegenerative disorders such as multiplesclerosis (Faber & Korn, 1989; Jefferys, 1995), but examples of ephatic effectsunder normal conditions are rare (Korn & Faber, 1980; Kanda & Axelrad,1980). Nonetheless, all electronic, cellular activity generates extracellularelectric fields and so it is natural to ask if these fields have any relevance tothe functioning of the brain. Before we can begin to answer this question,however, we need to consider how best to model these fields.Our current objective is to better understand the forward problem ofmodeling the extracellular field of various regions of the brain from the un-derlying, neural activity and to develop an accurate and efficient methodfor modeling these fields. A full construction of the extracellular field, fromsingle neuron activity, is extremely difficult. For instance, to generate mi-crovolt potentials, as commonly detected by electroencephalograph (EEG)scalp recordings, requires the superposition of activity from a great numberof neurons. A simple estimate is that it takes a 6 cm patch of cortical tissue,containing around 6 × synchronously active neurons, to generate a de-tectable signal on the order of microvolts (Ebersole, 1997). Nonetheless, themicroscopic behavior, although too difficult to incorporate exactly, may actas a guide in developing more coarse-grained models (Srinivasan, 2006). Forinstance, field theories of thalamic and cortical activity, constrained by phys-iological parameters, have recently been developed, and have proven success-ful in quantitatively reproducing various EEG phenomena, evoked response3otentials, coherence functions and seizure dynamics, among others. (Jirsa& Haken, 1996; Robinson et al., 2001; Robinson et al., 2005).Neurons display a variety of complicated geometries, giving rise to anarray of current distributions that dynamically vary throughout the courseof an action potential and during the interspike interval. For local probesof individual neurons–for instance, by microelectrodes–the field generated bythe action potential dominates, particularly near the soma. However, it isthought that synaptic activity as well as longer lasting depolarization and hy-perpolarizations are mainly responsible for the electrical activity detected byEEG recordings (Nunez & Srinivasan, 2006). There are two primary reasonswhy the contribution of the action potential is thought to be negligible to thefields detected by EEGs. First, in general, the dendritic axes of pyramidalcells lie parallel to the cortical sheet which allows the contribution of the ex-tracellular fields of the dendrites to add constructively, whereas the relativeorientation of their axons are more varied, leading to a significantly reducedaxonal contribution. Second, due to the relatively brief time course of anaction potential, neurons would need to precisely synchronize their firing inorder to generate a significant contribution to the extracellular field.In the current study, we focus on the extracellular field of a single spikingcell, with the future intention of quantifying hypotheses such as those dis-cussed above on the importance of the action potential to the extracellularfield. We base our work upon a quantitatively accurate model of a pyra-midal cell which our lab has developed and use this model to ask questions4egarding the local extracellular field, for instance, the field generated by asingle neuron or a minicolumn of pyramidal neurons, and later address howour results are relevant to more distant, global recordings, such as EEGs.The dynamics of the extracellular field of a spiking neuron are rathercomplex. One would like to remove some of the complexity of analyzing theextracellular field of realistic neurons by identifying the essential features thatcharacterize the current distributions. With this intention, our approach is toperform a moment expansion about the current distribution of the cell and tostudy the resulting, dynamical moments. Moment expansions are routinelyused in molecular biology to aid in the calculation of Coulomb mediatedmolecular interactions where the full electrostatic charge density may bequite complicated. They have been used to clarify the possible interactionsbetween normal and alkylated DNA base pairs (Price et al., 1993), to modelligand binding and protein-protein interactions (Neves-Petersen & Petersen,2003), and to simulate charge transport in biological ion channels (Saraniti,Aboud, & Eisenberg, 2006), to name only a few applications. Our presentgoal is two-fold: to first show that the dynamical moments of a biologicallyrealistic neuron can be efficiently calculated and to then see what simplifyingfeatures emerge from such an analysis.Our current approach naturally leads to several fundamental questionswhich have not been sufficiently addressed: when is it justified to model theneuron by a dipole; is there a region of interest where the first few momentsprovide a useful approximation to the extracellular field; close to the cell, do5ny of the moments dominate, or must we account for the full complexity ofthe current distribution? We present a method that is able to accurately andefficiently decompose the extracellular field into its fundamental momentsat all distances from the cell body. We then discuss the usefulness of suchan approach in describing local and global extracellular fields generated bynetworks of neurons. We begin by writing an equation for the extracellular field of a continuoussource of currents within the point-source approximation φ ( x ) = 14 πσ (cid:90) d x (cid:48) i ( x (cid:48) ) | x − x (cid:48) | , (1)where i ( x (cid:48) ) is the current at location x (cid:48) and x − x (cid:48) defines a vector whichpoints from the current source toward a test point at x . We will assume thatthe extracellular medium may be approximated as an homogeneous, isotropicvolume conductor and, therefore, the bulk conductivity tensor σ , may betaken as a constant. For frequency ranges between roughly 1 − −
400 Ω · cm.Since we are interested in a multipole expansion of the cell’s current dis-tribution at all distances from the cell, we need to pay particular attentionto the convergence properties of our expansion method. The usual decompo-sition into multipoles is based upon the following expansion of the 1 / | x − x (cid:48) | dependence of the electric potential into radial components r and sphericalharmonics Y l,m ( θ, φ ):1 | x − x (cid:48) | = ∞ (cid:88) l =0 l (cid:88) m = − l l + 1 r l< r l +1 > Y ∗ l,m ( θ (cid:48) , ϕ (cid:48) ) Y l,m ( θ, ϕ ) . (2)The symbol r < refers to the smaller of the two values of | x | and | x (cid:48) | (forinstance, x may be the vector which points to the test point while x (cid:48) pointsto the current source), while r > refers to the greater value. This conditionwill insure that the sum is convergent, so special care needs to be takento abide by this criterion. The classical multipole expansion assumes thatwe are outside the range of the current distribution, so we may identify r < with the magnitude of the vector pointing at current source r (cid:48) , while r > isassociated with a test point at r . However, due to the complicated geometrydisplayed by different neurons, we may easily find ourselves within a regimein which the identities of these two quantities are swapped.Figure (1) clarifies this point. We divide the extracellular region of astereotypical pyramidal cell into 2 distinct volumes. For convenience, wepick a point roughly halfway up the apical dendrite of the cell as our origin;7igure 1: The figure depicts the various regions into which the generalizedmultipole expansion is divided. Vector r’ points to the current source whilethe vector r is directed toward the test point. The solid circle divides region R from R . The dashed line marks the divide between the inner (inside)and outer (outside) field regions. The pyramidal cell depicted is purely illus-trative.however, this choice is arbitrary–for instance, we could have chosen the originto fall within the soma. Our choice of origin simply minimizes the totalspherical volume of the current containing region, which will later aid inthe numerics. For a test point at r , the region R denotes the volume overwhich r < = r (cid:48) , r > = r while region R is the volume where r < = r, r > = r (cid:48) .The solid line separates these two regions. It’s clear that for any value of r where there still exists an element of current outside the volume enclosedby that vector, we need to be careful that we have properly identified r < and r > . This leads to a natural splitting of extracellular space into tworegions, which is denoted by the dashed line in the figure. We will refer to8 test point within the volume enclosed by the dashed line as in the “inner-field,” while points outside will be considered the “outer-field.” We employthis terminology since the regions we are considering are somewhat differentthan the more typically encountered “near” and “far” field. The importantdistinction between this definition of an inner and outer-field is that theouter-field defines the region in which { r < , r > } are static whereas, withinthe inner-field, { r < , r > } vary based on the placement of the test point. Forinstance, for scalp recordings several centimeters from the relevant cells, oneis within the outer-field, but for intracranial recordings millimeters from acortical microcolumn, one might have to account for the inner-field based onthe position of the electrode.We may now write the following moment expansion of the extracellularpotential: φ ( x ) = 1 σ ∞ (cid:88) l =0 l (cid:88) m = − l Y l,m ( θ, ϕ ) (cid:18) q l,m r l +1 + r l p l,m (cid:19) , (3)where q l,m = (cid:90) R d x (cid:48) i ( x (cid:48) ) r (cid:48) l Y ∗ l,m ( θ (cid:48) , ϕ (cid:48) ) (4) p l,m = (cid:90) R d x (cid:48) i ( x (cid:48) ) r (cid:48) l +1 Y ∗ l,m ( θ (cid:48) , ϕ (cid:48) ) . (5)Equations (4,5) are the moments of the potential, q l,m are the classical multi-pole moments, while p l,m are the less familiar inverse moments. If we write theelements of the multipole expansion as φ l,m ( x ) = Y l,m ( θ, ϕ )( φ q l,m ( r ) + φ p l,m ( r )),9rom Eq. (3) we may define the classical and inverse radial potentials φ q l,m ( r ) ≡ σ q l,m r l +1 and φ p l,m ( r ) ≡ σ p l,m r l , (6)respectively. The radial potentials will be helpful in comparing the relativeimportance of the moments in the multipole expansion. Note the radialdependencies in Eq. (3) that guarantee convergence of the expansion. Inthe outer-field, this simply reduces to the standard multipole expansion, butthe series remains convergent within the inner-field as well, so long as werestrict our integration over the appropriate volume elements as denoted inEqs. (4,5) and illustrated in Fig. (1). A similar approach has recently beenused to study the electrostatic potential of topological atoms, from which wehave borrowed some of our terminology (Rafat & Popelier, 2005). The cell that we will work with is a biologically realistic model of a hip-pocampal pyramidal cell within rat CA1. The model was developed in (Goldet al., 2006) to compare intracellular recordings to simultaneous extracellularrecordings of neural activity. The active ionic currents were modeled usingHodgkin-Huxley style kinetics. Voltage dependent currents were carried byNa + , K + , and Ca ions and were modeled for 12 different current processes.Details of the model can be found in (Gold et al., 2006). To calculate the10 ! ! t (ms) I ( n A ) Figure 2: A representative time course of the total current across the somashowing the rapid inward (negative) Na + current, leading to the peak inthe action potential, and the slower, outward (positive) K + current whichrepolarizes the cell. Simulated synaptic input occurs within the first 1 mstriggering the firing of an action potential.extracellular field, we first computed the transmembrane currents for theneuron along with their associated ionic currents. Standard 1-D compart-mental simulations where performed within the NEURON Simulation En-vironment (Hines & Carnevale, 1997). Approximately 1000 compartmentswhere used to model an anatomically correct 3-D reconstruction of the cell.Within the first 1ms of the simulation we artifically depolarize the celluntil an action potential is triggered within the soma; the cell dynamicsfollow the course of the action potential until the cell repolarizes and returns11igure 3: Equipotential curves taken at the peak of the action potential, cal-culated from the original pyramidal cell, illustrating the approximate cylin-drical symmetry of the extracellular potential. We plot two cases above: a.)a plane at z = 250 µm , within the apical dendrites and b.) a plane at z = 0which is the location of the soma.to a stable resting potential. This choice of initiating the action potential isarbitrary, we could likewise apply the procedure discussed here to a cell whosefiring is initiated by synaptic input. Figure (2) shows the time course of themembrane current across a representative segment of the soma. Throughout,we assume that the extracellular potential is constant and equal to zero.We also assume that the transmembrane currents are not influenced by theevolving extracellular potentials ( (cid:28) !"" " !"" ! ! $"" ! %"" ! !"" ! &"""&""!""%""$"" ! !"" " !"" ! ! $"" ! %"" ! !"" ! &"""&""!""%""$"" µ m µ m µ m µ m a.) b.) Figure 4: a.) A projection of the pyramidal neuron onto a plane perpendic-ular to the cortical section (left panel). b.) The original cell is symmetrizedto simplify the analysis (right panel).that any anisotropy coming from the branched structure of the dendrites isunimportant. Assuming cylindrical symmetry allows us to reduce the dimen-sionality of the system from a three-dimensional calculation to a problem ofonly two-dimensions, but should only modify the quantitative, as opposed toqualitative, aspects of our results.We first project the neuron upon a plane parallel to the long axis of thecell body (Fig. (4a)). For a cortical pyramidal cell, the view would correspondto having flattened out the cortex and then looking at the cell in plane ofthe cortical sheet with the axon and basal dendrites toward the bottom andthe apical and distal dendrites reaching upward. Each point in the figure13orresponds to a current segment in the full, multi-compartmental model ofthis cell. It’s clear that the cell is not completely symmetric since the leftand right portions, relative to the vertical axis of the cell, do not exactlycorrespond. However, we neglect this anisotropy, and simply mirror the cellalong this axis, averaging any overlapping current segments (Fig. (4b)). Afterperforming this simple transformation, we now assume cylindrical symmetryalong the vertical axis of the cell, with the current elements providing acurrent density over the corresponding cylindrical volume. Viewed out ofplane, the cell would appear as an assortment of cylindrical annuli.By symmetrizing the cell, we greatly simplify the problem, since all termswhere m (cid:54) = 0 integrate to zero in Eq. (3). The remaining m = 0 sphericalharmonics are related to the Laguerre polynomials, P l ( x ), via Y l, ( θ, ϕ ) = (cid:113) (2 l + 1) / (4 π ) P l (cos θ ) which simplifies Eq. (3) and the calculation of themoments in Eqs. (4,5). We begin our analysis by considering the near-field moments which are rel-evant to local intracranial recordings of neural activity . In the inner-field,because of the changing volumes of the regions defined by R and R , bothclassical and inverse moments are dependent upon distance. To efficientlycompute the multipole expansion within this domain, we follow a similarprocedure to that outlined in (Rafat & Popelier, 2005). The idea is to divide14he inner-field into a series of N spherical shells and to then calculate theclassical and inverse moments in a piecewise fashion within each shell. Thiscalculation needs to be performed only once at each time step and may thenbe stored within a lookup table. To calculate the extracellular field of the cellrequires the evaluation of the integrals in Eqs.(4,5) which now become sumsover the appropriate subset of N shells, with an interpolation performed atthe boundary between regions R and R . Since the brunt of the numericsmay be performed ahead of time and stored within computer memory, thismethod provides an efficient way of calculating the moments at any radialdistance within the inner-field granted that the expansion converges for amodest number of terms. For the model pyramidal cell that we investigate,as an example, we take N=200 shells recorded over 200 time-steps. To storethe first 25 inverse and classical moments, we must generate a lookup tableof approximately 16 Mbytes which can easily be stored in the memory of amodern desktop computer.One might hope that only the first few moments define the extracellularfield of the cell; however, within the near field, the current distribution is toocomplex to allow such a simplification and the moment expansion containsmany comparable terms throughout the time course of the action potential.We illustrate this in Fig. (5) for a representative time ( t = 1 . φ q l ( r ) and 11 inverse φ p l ( r ) radial potentials.The fairly slow convergence of the weights of the expansion, displayed15 " ! $" ! ! ! !"! ! ! !"! radial distance ( µ m) φ q l ( r )( µ V ) radial distance ( µ m) φ p l ( r )( µ V ) Classical MomentsInverse Moments l = 1 l = 2 l = 0 l = 1 l = 2 l = 0 Figure 5: The radial potentials φ q ( r ) for the first 11 classical moments asa function of radial distance at t = 1 . l = 0 monopole moment (dash-dotted line), l = 1dipole (solid line), and the l = 2 quadrupole (dashed line) are emphasizedalong side the remaining moments up to l = 10 (dotted lines). Insert: Theradial potential φ p ( r ) for the first 11 inverse moments (same line labels asbefore) .in Fig. (5), is similar for various times about the action potential. If weexclude a radius of 10 − µ m about the center of the cell, guaranteeingwe are outside the body of the cell itself, the first 25 classical and inversemoments are needed to account for the total potential to within a few percentthroughout the entire timecourse of the action potential.From Fig. (5) it is hard to justify any dominant moments of the cellularcurrent distribution due to the strong radial dependence displayed. This16 " ! !%& ! !%!’!!%!’!%&!%&’!%"!%"’ ! " ! ’!’&!&’ ()&! ! * ! " ! ’!’&!&’"! ()&! ! t (ms) t (ms) t (ms) φ q l ( r )( µ V ) r = 0 . r = 1mm r = 5mm l = 2 l = 1 Figure 6: The radial potential φ q ( r ) from the first 11 classical moments,as a function of time, at various distances from the cell origin. The l = 1dipole moment (solid line) and the l = 2 quadrupole moment (dashed line)are emphasized. For comparison, the higher moments (dotted lines), up to l = 10, are shown.clearly implies that if we were to model the extracellular field of this neuronwithin radial distances on the order of half the length of the cell ( ∼ µ m),we must account for the full complexity of the current distribution, and thatany assumption of treating such a complex current distribution as, perhaps anoscillating dipole, would be unjustified. Nonetheless, summing over roughly50 elements (25 inverse moments and 25 classical moments) is a much quickerway to evaluate the extracellular field than summing over the ∼ Outer-Field cellular moments
Within the outer-field, our problem simplifies to a moment calculation whichcan be performed without any of the complexities introduced within theinner-field. One might assume, since the total current across the single neu-ron is conserved (i.e., the l = 0 moment is zero), that far from the cell the onlysignificant contributions would come from the dipole moment ( l = 1). How-ever, the quadrupole moment scales only one inverse power of r faster (1 /r as compared to 1 /r ). If we compare the magnitude of the 1 /r dipole po-tential to the 1 /r quadrupole potential at a point on the boundary betweenthe inner and outer-field ( r ∼ . ∼ l = 1 (solid line)and quadrupole l = 2 (dashed line) contributions to the radial potential φ q ( r ). The insets show the absolute value of the resulting total potential in µ V. Excluded is the inner-field ( r < µ m) where the cell would be orientedalong it’s vertical axis as shown in Fig (4). Left to right, starting on the top,the times are given by 1 . , . , . . r ∼ . r ∼ . . As previously shown, these twomoments dominate the extracellular field in the outer-field throughout theaction potential, except for at points close to the boundary. We, therefore,neglect the contributions of all higher moments ( l >
2) in the figures. Atapproximately 1 ms into the simulation, the cell begins to spike, and a largedipole moment dominates. However, as the action potential grows, a signifi-cant quadrupole moment emerges. The initial magnitude of this moment ismore than three times that of the dipole which means that it will contributeto the extracellular field over a significant spatial extent. When the actionpotential has peaked, the dipole moment has again gained in magnitude, andthe extracellular field is clearly dominated by this moment, which remainsuntil the hyperpolarization of the cell overshoots the threshold and a rel-atively strong quadrupole emerges again, although the overall extracellularpotential is much smaller at this point.
We have shown that the extracellular field of a biologically realistic pyramidalcell can be accurately and efficiently calculated at all spatial distances fromthe cell through a moment expansion of the membrane current distribution.We have formulated the multipole expansion in a form that converges at all High-resolution color images and animations of the extracellular potentials can befound at . r > . r > m = − l . . . l axial moments.This would likely lead to a large number of terms in the potential expansion,making the present procedure impractical. Second, we have treated the ex-21racellular medium as homogeneous, neglecting the effects of other dendritesor axons present within the vicinity of the cell. It would be very difficultif not impossible to exactly account for these inhomogeneities, nonetheless,it is an interesting question to ask, for instance, how random defects in theextracellular mileu might modulate the extracellular field. Third, we havetriggered the action potential within the soma and have analysed the extra-cellular field generated by the dynamics of the resulting membrane currents.One may also initiate the action potential by distributing the imputs withinthe synapses and proceed with the analysis we have presented here. Sincethe present method will work for an arbitrary current distribution, only theefficiency of our method should be effected.As discussed in the introduction, the contribution of the action potentialis thought to be negligible to EEG measurements. We are now in a betterposition to test this fundamental assumption. For instance, we may use themethod presented here to simulate large populations of biologically realis-tic spiking neurons and see the effects of orientation and synchrony on thecombined extracellular potentials. In particular, we may study the contribu-tions of slower components following the action potential, such as short andlonger-lasting after-hyperpolarizations. Unfortunately, due to the complex-ity of the current dynamics displayed by the model neuron we have used forthis study, it is difficult to infer how these slower processes would effect theextracellular fields without fully simulating these fields.22e would like to thank Carl Gold for providing us with his NEURON codepackage. Joshua Milstein and Christof Koch acknowledge support from theSwartz Foundation and NSF. References
Bedard, C., Kroger, H., & Destexthe, A. (2004). Modeling extracellularfield potentials and the frequency filtering properties of extracellular space.
Biophysical J. , 86, 1829-1842.Ebersole, J. S., (1997). Defining epileptogenic foci: past, present, future.
J.Clin. Neurophys. , 14, 470-483.Faber, D. S., & Korn, H. (1989). Electrical field effects: Their relevance incentral neural networks.
Physiol. Rev. , 69,821-863.Gold, C., Henze, D. A., Koch, C., & Buzsaki, G. (2006). On the Origin ofthe Extracellular Action Potential Waveform: A modeling study.
Journalof Neurophysiology , 95, 3113-3128.Henze, D. A., Borhegyi, Z., Csicsvari, J., Mamiya, A., Harris, K. D.,& Buzsaki, G. (2000). Intracellular Features Predicted by ExtracellularRecordings in the Hippicampus in Vivo.
Journal of Neurophysiology , 84,390-400. 23ines, M. L., & Carnevale, N. T. (1997). The neuron simulation environ-ment.
Neural Comput. , 9, 1179-1209.Hodgkin, A. L., & Huxley A. F. (1952d). A quantitative description ofmembrane current and its application to conduction and excitation in nerve.
J. Physiol. , 117,500-544.Holt G. R., & Koch C. (1999). Electrical Interactions via the ExtracellularPotential Near Cell Bodies.
Journal of Computational Neuroscience , 6, 169-184.Jackson, J. D. (1975).
Classical Electrodynamics.
New York: John Wiley &Sons.Jefferys, J. G. R. (1995). Nonsynaptic modulation of neuronal activity inthe brain: Electric current and extracellular ions.
Physiol. Rev. , 75, 689-723.Jirsa, V. K., & Haken, H. (1996). Field Theory of Electromagnetic BrainActivity.
Phys. Rev. Lett. , 77, 960-963.Koch, C. (1999).
Biophysics of Computation . New York: Oxford UniversityPress.Korn, H., & Axelrad, H. (1980). Electrical inhibition of Purkinje cells in thecerebellum of the rat.
Proc. Natl. Acad. Sci. , 77, 6244-6247.Korn, H., & Faber, D. S. (1980). Electrical field effect interactions in thevertebrate brain.
Trends Neurosci. , 3, 6-9.24eves-Petersen, M. T., & Petersen, S. B. (2003). Protein electrostatics: areview of the equations and methods used to model electrostatic equationsin biomolecules–applications in biotechnology.
Biotechnol. Annu. Rev. , 9,315-95.Nunez, P. L., & Srinivasan, R. (2006).
Electric Fields of the Brain . NewYork: Oxford University Press.Plonsey, R. (1969).
Bioelectric Phenomena . New York: McGraw-Hill.Price, S. L., Celso, F. L., Treichel, J. A., Goodfellow, J. M., & Umrania,Y. (1993). What Base Pairings Can Occur in DNA? A Distributed Multi-pole Study of the Electrostatic Interactions between Normal and AlkylatedNucleic Acid Bases.
J. Chem. Soc. , 89, 3407-3417.Rafat, M., & Popelier, P. L. A. (2005). The electrostatic potential gener-ated by topological atoms. II. Inverse multipole moments.
The Journal ofChemical Physics , 123, 204103, 1-7.Robinson, P. A., Rennie, C. J., Rowe, D. L., O’Conner, S. C., & Gordon, E.(2005). Multiscale brain modeling.
Phi. Trans. Roy. Soc. Lon. B , 360,1043-1050.Robinson, P. A., Rennie, C. J., Wright, J. J., Bahramali, H., Gordon, E.,& Rowe, D. L. (2001). Prediction of electroencephalographic spectra fromneurophysiology.
Phys. Rev. E. , 63, 021903, 1-1825araniti, M., Aboud, S., & Eisenberg, R. (2006). The Simulation of IonicCharge Transport in Biological Ion Channels: An Introduction to NumericalMethods.
Reviews in Computational Chemistry , 22, 229-293.Srinivasan, R. (2006). Anatomical Constraints on Source Models for High-resolution EEG and MEG Derived from MRI.