Consider the situation in which a small electrode of radius a is placed in a homogeneous spherical region of radius R filled with uniformly distributed neural generators. One goal of this paper will be to compute the cross covariance function: ν_aR (x→ MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacuWG4baEgaWcaaaa@2E37@, s→ MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacuWGZbWCgaWcaaaa@2E2D@, τ) which represents the cross covariance between signals recorded from two electrodes one located at position x→ MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacuWG4baEgaWcaaaa@2E37@ and another located a distance s→ MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacuWGZbWCgaWcaaaa@2E2D@ away from the first as a function of a, R, τ(time difference) and the assumptions about the correlation between neural generators throughout the medium. Also of interest will be a study of ν_aR (τ) = ν_aR (0,0, τ) which is the autocovariance function for signals recorded from the center of the spherical medium as a function of R. Situations in which the value of lim⁡R→∞νaR(τ) MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaadaWfqaqaaiGbcYgaSjabcMgaPjabc2gaTbWcbaGaemOuaiLaeyOKH4QaeyOhIukabeaaiiGakiab=17aUnaaBaaaleaacqWGHbqycqWGsbGuaeqaaOGaeiikaGIae8hXdqNaeiykaKcaaa@3D77@ does not exist suggest that under these conditions, recordings will be dominated by contributions from distant neurons and will be strongly dependent on the size and shape of the region in which the electrode is immersed. The core of the problem can be easily illustrated in a simple argument reminiscent of that which leads to Olber's paradox. If the potential from a set of neural sources a distance r from an electrode falls as 1rm MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaadaWcaaqaaiabigdaXaqaaiabdkhaYnaaCaaaleqabaGaemyBa0gaaaaaaaa@30A9@ the net contribution to the potential from all neural sources in a spherical shell located at distances between R and R + ΔR from the electrode will be proportional to 4πR2ΔR1Rm MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacqaI0aaniiGacqWFapaCcqWGsbGudaahaaWcbeqaaiabikdaYaaakiabgs5aejabdkfasnaalaaabaGaeGymaedabaGaemOuai1aaWbaaSqabeaacqWGTbqBaaaaaaaa@380D@. If m is less than or equal to 2 each successively more distant shell of sources contributes either a greater or equal amount to the total potential and hence the recorded potential is very sensitive to the exact shape and size of the neural region. When m = 3, the contribution from each successively more distant shell diminishes as 1R MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaadaWcaaqaaiabigdaXaqaaiabdkfasbaaaaa@2ED9@ but the total potential diverges logarithmically and so, even in this case, it is expected that the recorded potential will be strongly dependent on the shape and size of the volume in which the recording electrode is placed. Only when m > 3 does the measured potential reach a finite limit for large values of R and hence most of the electrical activity recorded can be considered to come from neurons near the electrode. This result suggests that when recording from highly correlated dipolar or quadrupolar sources most of the electrical activity recorded from an electrode comes from neurons far from the electrode. This paradoxical result arises out of the implicit assumption that neuronal activity is highly correlated even at large distances. The following discussion will demonstrate how the above result depends on the details of the correlation function describing activity in spatially separated neurons.

The general expression for the electric potential produced at the location of the electrode x→ MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacuWG4baEgaWcaaaa@2E37@ is:

ϕ ( x → , t ) = ∫ V d 3 x → ′ ρ ( x → ′ , t ) | x → − x → ′ |       ( 1 ) MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaaiiGacqWFvpGAcqGGOaakcuWG4baEgaWcaiabcYcaSiabdsha0jabcMcaPiabg2da9maapefabaGaemizaq2aaWbaaSqabeaacqaIZaWmaaGccuWG4baEgaWcgaqbaaWcbaGaemOvayfabeqdcqGHRiI8aOWaaSaaaeaacqWFbpGCcqGGOaakcuWG4baEgaWcgaqbaiabcYcaSiabdsha0jabcMcaPaqaaiabcYha8jqbdIha4zaalaGaeyOeI0IafmiEaGNbaSGbauaacqGG8baFaaGaaCzcaiaaxMaadaqadaqaaiabigdaXaGaayjkaiaawMcaaaaa@4EDD@

where V is the spherical volume of radius R and ρ(x→ MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacuWG4baEgaWcaaaa@2E37@', t) is the charge density at the point x→ MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacuWG4baEgaWcaaaa@2E37@' and time t. Consider the contribution to the total potential made by the generators in a small element of space ΔVx→1 MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacqGHuoarcqWGwbGvdaWgaaWcbaGafmiEaGNbaSaadaWgaaadbaGaeGymaedabeaaaSqabaaaaa@3227@ centered around the point x→ MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacuWG4baEgaWcaaaa@2E37@₁. This is given by:

ϕ ( x → , t , Δ V x → 1 ) = ∫ Δ V x → 1 d 3 x → ″ ρ ( x → 1 + x → ″ , t ) | x → − x → 1 − x → ″ |       ( 2 ) MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaaiiGacqWFvpGAcqGGOaakcuWG4baEgaWcaiabcYcaSiabdsha0jabcYcaSiabgs5aejabdAfawnaaBaaaleaacuWG4baEgaWcamaaBaaameaacqaIXaqmaeqaaaWcbeaakiabcMcaPiabg2da9maapefabaGaemizaq2aaWbaaSqabeaacqaIZaWmaaGccuWG4baEgaWcgaGbaaWcbaGaeyiLdqKaemOvay1aaSbaaWqaaiqbdIha4zaalaWaaSbaaeaacqaIXaqmaeqaaaqabaaaleqaniabgUIiYdGcdaWcaaqaaiab=f8aYjabcIcaOiqbdIha4zaalaWaaSbaaSqaaiabigdaXaqabaGccqGHRaWkcuWG4baEgaWcgaGbaiabcYcaSiabdsha0jabcMcaPaqaaiabcYha8jqbdIha4zaalaGaeyOeI0IafmiEaGNbaSaadaWgaaWcbaGaeGymaedabeaakiabgkHiTiqbdIha4zaalyaagaGaeiiFaWhaaiaaxMaacaWLjaWaaeWaaeaacqaIYaGmaiaawIcacaGLPaaaaaa@60B3@

This expression can be rearranged to display the potential in terms of the multipole expansion of the charge density in the region by using the relation 3:

1 | x → − x → 1 − x → ″ | = 4 π ∑ l = 0 ∞ ∑ m = − l l 1 2 l + 1 | x → ″ | l | x → − x → 1 | l + 1 Y l m ∗ ( θ ″ , φ ″ ) Y l m ( θ 1 , φ 1 )       ( 3 ) MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaadaWcaaqaaiabigdaXaqaaiabcYha8jqbdIha4zaalaGaeyOeI0IafmiEaGNbaSaadaWgaaWcbaGaeGymaedabeaakiabgkHiTiqbdIha4zaalyaagaGaeiiFaWhaaiabg2da9iabisda0GGaciab=b8aWnaaqahabaWaaabCaeaadaWcaaqaaiabigdaXaqaaiabikdaYiabdYgaSjabgUcaRiabigdaXaaadaWcaaqaaiabcYha8jqbdIha4zaalyaagaGaeiiFaW3aaWbaaSqabeaacqWGSbaBaaaakeaacqGG8baFcuWG4baEgaWcaiabgkHiTiqbdIha4zaalaWaaSbaaSqaaiabigdaXaqabaGccqGG8baFdaahaaWcbeqaaiabdYgaSjabgUcaRiabigdaXaaaaaaabaGaemyBa0Maeyypa0JaeyOeI0IaemiBaWgabaGaemiBaWganiabggHiLdaaleaacqWGSbaBcqGH9aqpcqaIWaamaeaacqGHEisPa0GaeyyeIuoakiabdMfaznaaDaaaleaacqWGSbaBcqWGTbqBaeaacqGHxiIkaaGccqGGOaakcuWF4oqCgaGbaiabcYcaSiqb=z8aMzaagaGaeiykaKIaemywaK1aaSbaaSqaaiabdYgaSjabd2gaTbqabaGccqGGOaakcqWF4oqCdaWgaaWcbaGaeGymaedabeaakiabcYcaSiab=z8aMnaaBaaaleaacqaIXaqmaeqaaOGaeiykaKIaaCzcaiaaxMaadaqadaqaaiabiodaZaGaayjkaiaawMcaaaaa@7DD6@

where the angles (θ", φ") describe local coordinates (i.e. orientation of x→ MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacuWG4baEgaWcaaaa@2E37@") within ΔVx→1 MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacqGHuoarcqWGwbGvdaWgaaWcbaGafmiEaGNbaSaadaWgaaadbaGaeGymaedabeaaaSqabaaaaa@3227@, (θ₁, φ₁") describe the orientation of the region ΔVx→1 MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacqGHuoarcqWGwbGvdaWgaaWcbaGafmiEaGNbaSaadaWgaaadbaGaeGymaedabeaaaSqabaaaaa@3227@ in relation to the origin of coordinates, and it is assumed that the region of interest is smaller in its maximum diameter than the distance between the origin of coordinates and the region. The Y_lm (θ, φ) are spherical harmonics and Ylm∗ MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacqWGzbqwdaqhaaWcbaGaemiBaWMaemyBa0gabaGaey4fIOcaaaaa@31C7@ (θ, φ) are their complex conjugates. Substituting (3) into (2) demonstrates that the total contribution to the potential from sources in the region ΔVx→1 MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacqGHuoarcqWGwbGvdaWgaaWcbaGafmiEaGNbaSaadaWgaaadbaGaeGymaedabeaaaSqabaaaaa@3227@ can be written as:

ϕ ( x → , t , Δ V x → 1 ) = 4 π ∑ l = 0 ∞ ∑ m = − l l 1 2 l + 1 1 | x → − x → 1 | l + 1 Y l m ( θ 1 , φ 1 ) Q l m ( x → 1 , t ) Δ V x → 1       ( 4 ) MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaaiiGacqWFvpGAcqGGOaakcuWG4baEgaWcaiabcYcaSiabdsha0jabcYcaSiabgs5aejabdAfawnaaBaaaleaacuWG4baEgaWcamaaBaaameaacqaIXaqmaeqaaaWcbeaakiabcMcaPiabg2da9iabisda0iab=b8aWnaaqahabaWaaabCaeaadaWcaaqaaiabigdaXaqaaiabikdaYiabdYgaSjabgUcaRiabigdaXaaadaWcaaqaaiabigdaXaqaaiabcYha8jqbdIha4zaalaGaeyOeI0IafmiEaGNbaSaadaWgaaWcbaGaeGymaedabeaakiabcYha8naaCaaaleqabaGaemiBaWMaey4kaSIaeGymaedaaaaaaeaacqWGTbqBcqGH9aqpcqGHsislcqWGSbaBaeaacqWGSbaBa0GaeyyeIuoaaSqaaiabdYgaSjabg2da9iabicdaWaqaaiabg6HiLcqdcqGHris5aOGaemywaK1aaSbaaSqaaiabdYgaSjabd2gaTbqabaGccqGGOaakcqWF4oqCdaWgaaWcbaGaeGymaedabeaakiabcYcaSiab=z8aMnaaBaaaleaacqaIXaqmaeqaaOGaeiykaKIaemyuae1aaSbaaSqaaiabdYgaSjabd2gaTbqabaGccqGGOaakcuWG4baEgaWcamaaBaaaleaacqaIXaqmaeqaaOGaeiilaWIaemiDaqNaeiykaKIaeyiLdqKaemOvay1aaSbaaSqaaiqbdIha4zaalaWaaSbaaWqaaiabigdaXaqabaaaleqaaOGaaCzcaiaaxMaadaqadaqaaiabisda0aGaayjkaiaawMcaaaaa@7FCE@

where

Q l m ( x → 1 , t ) = 1 Δ V x → 1 ∫ Δ V x → 1 | x → ″ | l Y l m ∗ ( θ ″ , φ ″ ) ρ ( x → 1 + x → ″ , t ) d 3 x → ″       ( 5 ) MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacqWGrbqudaWgaaWcbaGaemiBaWMaemyBa0gabeaakiabcIcaOiqbdIha4zaalaWaaSbaaSqaaiabigdaXaqabaGccqGGSaalcqWG0baDcqGGPaqkcqGH9aqpdaWcaaqaaiabigdaXaqaaiabgs5aejabdAfawnaaBaaaleaacuWG4baEgaWcamaaBaaameaacqaIXaqmaeqaaaWcbeaaaaGcdaWdrbqaaiabcYha8bWcbaGaeyiLdqKaemOvay1aaSbaaWqaaiqbdIha4zaalaWaaSbaaeaacqaIXaqmaeqaaaqabaaaleqaniabgUIiYdGccuWG4baEgaWcgaGbaiabcYha8naaCaaaleqabaGaemiBaWgaaOGaemywaK1aa0baaSqaaiabdYgaSjabd2gaTbqaaiabgEHiQaaakiabcIcaOGGaciqb=H7aXzaagaGaeiilaWIaf8NXdyMbayaacqGGPaqkcqWFbpGCcqGGOaakcuWG4baEgaWcamaaBaaaleaacqaIXaqmaeqaaOGaey4kaSIafmiEaGNbaSGbayaacqGGSaalcqWG0baDcqGGPaqkcqWGKbazdaahaaWcbeqaaiabiodaZaaakiqbdIha4zaalyaagaGaaCzcaiaaxMaadaqadaqaaiabiwda1aGaayjkaiaawMcaaaaa@6B05@

Q_lm is the average multipole moment per unit volume at x→ MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacuWG4baEgaWcaaaa@2E37@₁ and time t (ie the multipole moment density). It is important to understand the appropriate choice of the volume. It should be chosen to be large enough that each volume consists of a very similar collection of neural elements but small enough that its leading multipole moments are similar each time the region is activated. This restriction can be relaxed greatly without altering the underlying conclusions of this study if one considers the fields generated by coupled neural structures of different types. The total potential will simply be the sum of the potentials generated by the structures of each type.

Integrating (4) over all the volume elements in the sphere system yields the following expression for the total recorded potential as:

ϕ ( x → , t ) = 4 π ∫ V ∑ l = 0 ∞ ∑ m = − l l 1 2 l + 1 1 | x → − x → 1 | l + 1 Y l m ( θ 1 , φ 1 ) Q l m ( x → 1 , t ) d 3 x → 1       ( 6 ) MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaaiiGacqWFvpGAcqGGOaakcuWG4baEgaWcaiabcYcaSiabdsha0jabcMcaPiabg2da9iabisda0iab=b8aWnaapefabaWaaabCaeaadaaeWbqaamaalaaabaGaeGymaedabaGaeGOmaiJaemiBaWMaey4kaSIaeGymaedaamaalaaabaGaeGymaedabaGaeiiFaWNafmiEaGNbaSaacqGHsislcuWG4baEgaWcamaaBaaaleaacqaIXaqmaeqaaOGaeiiFaW3aaWbaaSqabeaacqWGSbaBcqGHRaWkcqaIXaqmaaaaaaqaaiabd2gaTjabg2da9iabgkHiTiabdYgaSbqaaiabdYgaSbqdcqGHris5aaWcbaGaemiBaWMaeyypa0JaeGimaadabaGaeyOhIukaniabggHiLdGccqWGzbqwdaWgaaWcbaGaemiBaWMaemyBa0gabeaakiabcIcaOiab=H7aXnaaBaaaleaacqaIXaqmaeqaaOGaeiilaWIae8NXdy2aaSbaaSqaaiabigdaXaqabaGccqGGPaqkcqWGrbqudaWgaaWcbaGaemiBaWMaemyBa0gabeaakiabcIcaOiqbdIha4zaalaWaaSbaaSqaaiabigdaXaqabaGccqGGSaalcqWG0baDcqGGPaqkcqWGKbazdaahaaWcbeqaaiabiodaZaaakiqbdIha4zaalaWaaSbaaSqaaiabigdaXaqabaaabaGaemOvayfabeqdcqGHRiI8aOGaaCzcaiaaxMaadaqadaqaaiabiAda2aGaayjkaiaawMcaaaaa@7C89@

where (θ₁, φ₁) are the angles describing the location of the element under study relative to the center of the coordinate system. The advantage of this expansion over (1) is that in many neural systems the fields generated are dominated by either quadrupolar (l = 2) or dipolar (l = 1) components and so only theses components of the above sum are of practical importance.

One of the most commonly used descriptors of spontaneous electrical activity is the cross-covariance function. The goal of this section is to relate the patterns of multipole moment activation to the covariance between the signal recorded by an electrode of radius a and a similar electrode a distance s→ MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacuWGZbWCgaWcaaaa@2E2D@ from the first electrode embedded in spherical volume of radius R. One standard definition of the cross covariance function is:

ν a R ( x → , s → , τ ) = 〈 ( ϕ ( x → , t ) − 〈 ϕ ( x → , t ) 〉 ) ( ϕ ( x → + s → , t + τ ) − 〈 ϕ ( x → + s → , t + τ ) 〉 ) 〉       ( 7 ) MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaaiiGacqWF9oGBdaWgaaWcbaGaemyyaeMaemOuaifabeaakiabcIcaOiqbdIha4zaalaGaeiilaWIafm4CamNbaSaacqGGSaalcqWFepaDcqGGPaqkcqGH9aqpdaaadeqaamaabmaabaGae8x1dOMaeiikaGIafmiEaGNbaSaacqGGSaalcqWG0baDcqGGPaqkcqGHsisldaaadeqaaiab=v9aQjabcIcaOiqbdIha4zaalaGaeiilaWIaemiDaqNaeiykaKcacaGLPmIaayPkJaaacaGLOaGaayzkaaWaaeWaaeaacqWFvpGAcqGGOaakcuWG4baEgaWcaiabgUcaRiqbdohaZzaalaGaeiilaWIaemiDaqNaey4kaSIae8hXdqNaeiykaKIaeyOeI0YaaaWabeaacqWFvpGAcqGGOaakcuWG4baEgaWcaiabgUcaRiqbdohaZzaalaGaeiilaWIaemiDaqNaey4kaSIae8hXdqNaeiykaKcacaGLPmIaayPkJaaacaGLOaGaayzkaaaacaGLPmIaayPkJaGaaCzcaiaaxMaadaqadaqaaiabiEda3aGaayjkaiaawMcaaaaa@6FE3@

where the following notation is used:

〈 f ( t ) 〉 = lim ⁡ T → ∞ 1 2 T ∫ − T T f ( t ) d t       ( 8 ) MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaadaaadeqaaiabdAgaMjabcIcaOiabdsha0jabcMcaPaGaayzkJiaawQYiaiabg2da9maaxababaGagiiBaWMaeiyAaKMaeiyBa0galeaacqWGubavcqGHsgIRcqGHEisPaeqaaOWaaSaaaeaacqaIXaqmaeaacqaIYaGmcqWGubavaaWaa8qCaeaacqWGMbGzcqGGOaakcqWG0baDcqGGPaqkcqWGKbazcqWG0baDaSqaaiabgkHiTiabdsfaubqaaiabdsfaubqdcqGHRiI8aOGaaCzcaiaaxMaadaqadaqaaiabiIda4aGaayjkaiaawMcaaaaa@50D5@

for any function of time f(t). Substituting (6) into (7) yields:

ν a R ( x → , s → , τ ) ( 4 π ) 2 = ∫ V d 3 x → ′ 1 d 3 x → 1 ∑ l = 0 ∞ ∑ l ′ = 0 ∞ ∑ m = − l l ∑ m ′ = − l ′ l ′ 1 2 l + 1 1 2 l ′ + 1 1 | x → − x → 1 | l + 1 Y l m ( θ 1 , φ 1 ) Y l ′ m ′ ( θ ′ 1 , φ ′ 1 ) | x → + s → − x → ′ 1 | l ′ + 1 〈 [ Q l m ( x → 1 , t ) − 〈 Q l m ( x → 1 , t ) 〉 ] [ Q l ′ m ′ ( x → ′ 1 , t + τ ) − 〈 Q l ′ m ′ ( x → ′ 1 , t + τ ) 〉 ] 〉       ( 9 ) MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=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z8aMnaaBaaaleaacqaIXaqmaeqaaOGaeiykaKIaemywaK1aaSbaaSqaaiqbdYgaSzaafaGafmyBa0MbauaaaeqaaOGaeiikaGIaf8hUdeNbauaadaWgaaWcbaGaeGymaedabeaakiabcYcaSiqb=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@EB15@

where V is now taken as the region bounded by the surface of both electrodes and the sphere of radius R which bounds the region filled with neural structures.

One very important simplification occurs when it can be assumed that the structure of each neural region remains constant over time and that the fluctuations in neural activity over time in each region just modify the magnitude of the multipole moments over time. In particular, this means that it is possible to write:

Q l m ( x → 1 , t ) = ℚ l m ( x → 1 ) σ ( x → 1 , t )       ( 10 ) MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacqWGrbqudaWgaaWcbaGaemiBaWMaemyBa0gabeaakiabcIcaOiqbdIha4zaalaWaaSbaaSqaaiabigdaXaqabaGccqGGSaalcqWG0baDcqGGPaqkcqGH9aqpcqWIAecPdaWgaaWcbaGaemiBaWMaemyBa0gabeaakiabcIcaOiqbdIha4zaalaWaaSbaaSqaaiabigdaXaqabaGccqGGPaqkiiGacqWFdpWCcqGGOaakcuWG4baEgaWcamaaBaaaleaacqaIXaqmaeqaaOGaeiilaWIaemiDaqNaeiykaKIaaCzcaiaaxMaadaqadaqaaiabigdaXiabicdaWaGaayjkaiaawMcaaaaa@4E81@

where ℚ_lm(x→ MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacuWG4baEgaWcaaaa@2E37@₁) gives the magnitude of the maximal multipole moment generated when each region is fully activated and σ(x→ MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacuWG4baEgaWcaaaa@2E37@₁, t) is a scalar function describing the degree of activation of neural elements at position x→ MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacuWG4baEgaWcaaaa@2E37@₁ at time t. Note that in this representation, the orientation of a specific multipole moments is specified by the relative weights of the moment for the different values of m for a given value of 1 and so this theory also allows for arbitrary variations of the multipole moments from location to location. This means that:

〈 [ Q l m ( x → 1 , t ) − 〈 Q l m ( x → 1 , t ) 〉 ] [ Q l ′ m ′ ( x → 1 ′ , t + τ ) − 〈 Q l ′ m ′ ( x → 1 ′ , t + τ ) 〉 ] 〉 = ℚ l m ( x → 1 ) ℚ l ′ m ′ ( x → 1 ′ ) 〈 [ σ ( x → 1 , t ) − 〈 σ ( x → 1 , t ) 〉 ] [ σ ( x → 1 ′ , t + τ ) − 〈 σ ( x → 1 ′ , t + τ ) 〉 ] 〉       ( 11 ) MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=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@B9CE@

Substituting into (10) yields:

ν a R ( x → , s → , τ ) ( 4 π ) 2 = ∫ V d 3 x → ′ 1 d 3 x → 1 ∑ l = 0 ∞ ∑ l ′ = 0 ∞ ∑ m = − l l ∑ m ′ = − l ′ l ′ 1 2 l + 1 1 2 l ′ + 1 Y l m ( θ 1 , φ 1 ) Y l ′ m ′ ( θ ′ 1 , φ ′ 1 ) | x → − x → 1 | l + 1 ℚ l m ( x → 1 ) ℚ l ′ m ′ ( x → 1 ′ ) | x → + s → − x → ′ 1 | l ′ + 1 g ( x → 1 − x → 1 ′ , x → 1 , τ )       ( 12 ) MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=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@D2F6@

where:

g ( x → 1 − x → 1 ′ , x → 1 , τ ) = 〈 [ σ ( x → 1 , t ) − 〈 σ ( x → 1 , t ) 〉 ] [ σ ( x → 1 ′ , t + τ ) − 〈 σ ( x → 1 ′ , t + τ ) 〉 ] 〉       ( 13 ) MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacqWGNbWzcqGGOaakcuWG4baEgaWcamaaBaaaleaacqaIXaqmaeqaaOGaeyOeI0IafmiEaGNbaSaadaWgaaWcbaGaeGymaedabeaaiiaakiab=jdiIkabcYcaSiqbdIha4zaalaWaaSbaaSqaaiabigdaXaqabaGccqGGSaaliiGacqGFepaDcqGGPaqkcqGH9aqpdaaadeqaamaadmaabaGae43WdmNaeiikaGIafmiEaGNbaSaadaWgaaWcbaGaeGymaedabeaakiabcYcaSiabdsha0jabcMcaPiabgkHiTmaaamqabaGae43WdmNaeiikaGIafmiEaGNbaSaadaWgaaWcbaGaeGymaedabeaakiabcYcaSiabdsha0jabcMcaPaGaayzkJiaawQYiaaGaay5waiaaw2faamaadmaabaGae43WdmNaeiikaGIafmiEaGNbaSaadaWgaaWcbaGaeGymaedabeaakiab=jdiIkabcYcaSiabdsha0jabgUcaRiab+r8a0jabcMcaPiabgkHiTmaaamqabaGae43WdmNaeiikaGIafmiEaGNbaSaadaWgaaWcbaGaeGymaedabeaakiab=jdiIkabcYcaSiabdsha0jabgUcaRiab+r8a0jabcMcaPaGaayzkJiaawQYiaaGaay5waiaaw2faaaGaayzkJiaawQYiaiaaxMaacaWLjaWaaeWaaeaacqaIXaqmcqaIZaWmaiaawIcacaGLPaaaaaa@78A3@

is the paired covariance between the level of activity in different regions of the neural network. It should be noted that the covariance has been assumed to depend only on the distance between the regions and the location of one of the regions.

Before passing onto the more general case discussed in Appendix B, it is instructive to consider the case in which the level of activation at in each region is totally uncorrelatated:

g ( x → 1 − x → 1 ′ , x → 1 , τ ) = δ ( x → 1 − x → 1 ′ ) h ( x → 1 , τ )       ( 14 ) MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacqWGNbWzcqGGOaakcuWG4baEgaWcamaaBaaaleaacqaIXaqmaeqaaOGaeyOeI0IafmiEaGNbaSaadaWgaaWcbaGaeGymaedabeaaiiaakiab=jdiIkabcYcaSiqbdIha4zaalaWaaSbaaSqaaiabigdaXaqabaGccqGGSaaliiGacqGFepaDcqGGPaqkcqGH9aqpcqGF0oazcqGGOaakcuWG4baEgaWcamaaBaaaleaacqaIXaqmaeqaaOGaeyOeI0IafmiEaGNbaSaadaWgaaWcbaGaeGymaedabeaakiab=jdiIkabcMcaPiabdIgaOjabcIcaOiqbdIha4zaalaWaaSbaaSqaaiabigdaXaqabaGccqGGSaalcqGFepaDcqGGPaqkcaWLjaGaaCzcamaabmaabaGaeGymaeJaeGinaqdacaGLOaGaayzkaaaaaa@56F0@

for any function h(x→ MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacuWG4baEgaWcaaaa@2E37@₁, τ). This implies that:

ν a R ( x → , s → , τ ) ( 4 π ) 2 = ∫ V d 3 x → 1 ∑ l = 0 ∞ ∑ l ′ = 0 ∞ ∑ m = − l l ∑ m ′ = − l ′ l ′ 1 2 l + 1 1 2 l ′ + 1 1 | x → − x → 1 | l + 1 1 | x → − s → − x → 1 | l ′ + 1 Y l m ( θ 1 , φ 1 ) Y l ′ m ′ ( θ 1 , φ 1 ) ℚ l m ( x → 1 ) ℚ l ′ m ′ ( x → 1 ) h ( x → 1 , τ )       ( 15 ) MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=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H7aXnaaBaaaleaacqaIXaqmaeqaaOGaeiilaWIae8NXdy2aaSbaaSqaaiabigdaXaqabaGccqGGPaqkcqWIAecPdaWgaaWcbaGaemiBaWMaemyBa0gabeaakiabcIcaOiqbdIha4zaalaWaaSbaaSqaaiabigdaXaqabaGccqGGPaqkcqWIAecPdaWgaaWcbaGafmiBaWMbauaacuWGTbqBgaqbaaqabaGccqGGOaakcuWG4baEgaWcamaaBaaaleaacqaIXaqmaeqaaOGaeiykaKIaemiAaGMaeiikaGIafmiEaGNbaSaadaWgaaWcbaGaeGymaedabeaakiabcYcaSiab=r8a0jabcMcaPiaaxMaacaWLjaWaaeWaaeaacqaIXaqmcqaI1aqnaiaawIcacaGLPaaaaaaa@C553@

In Appendix D, the situation where s→ MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacuWGZbWCgaWcaaaa@2E2D@ is not zero is discussed, but in the case where both s→ MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacuWGZbWCgaWcaaaa@2E2D@ = x→ MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacuWG4baEgaWcaaaa@2E37@ = 0:

ν a R ( τ ) = ( 4 π ) 2 ∫ V d 3 x → 1 ∑ l = 0 ∞ ∑ l ′ = 0 ∞ ∑ m = − l l ∑ m ′ = − l ′ l ′ 1 2 l + 1 1 2 l ′ + 1 1 | x → 1 | l + l ′ + 2 Y l m ( θ 1 , φ 1 ) Y l ′ m ′ ( θ 1 , φ 1 ) ℚ l m ( x → 1 ) ℚ l ′ m ′ ( x → 1 ) h ( x → 1 , τ )       ( 16 ) MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaaiiGacqWF9oGBdaWgaaWcbaGaemyyaeMaemOuaifabeaakiabcIcaOiab=r8a0jabcMcaPiabg2da9iabcIcaOiabisda0iab=b8aWjabcMcaPmaaCaaaleqabaGaeGOmaidaaOWaa8quaeaacqWGKbazdaahaaWcbeqaaiabiodaZaaakiqbdIha4zaalaWaaSbaaSqaaiabigdaXaqabaGcdaaeWbqaamaaqahabaWaaabCaeaadaaeWbqaamaalaaabaGaeGymaedabaGaeGOmaiJaemiBaWMaey4kaSIaeGymaedaamaalaaabaGaeGymaedabaGaeGOmaiJafmiBaWMbauaacqGHRaWkcqaIXaqmaaWaaSaaaeaacqaIXaqmaeaacqGG8baFcuWG4baEgaWcamaaBaaaleaacqaIXaqmaeqaaOGaeiiFaW3aaWbaaSqabeaacqWGSbaBcqGHRaWkcuWGSbaBgaqbaiabgUcaRiabikdaYaaaaaaabaGafmyBa0MbauaacqGH9aqpcqGHsislcuWGSbaBgaqbaaqaaiqbdYgaSzaafaaaniabggHiLdaaleaacqWGTbqBcqGH9aqpcqGHsislcqWGSbaBaeaacqWGSbaBa0GaeyyeIuoaaSqaaiqbdYgaSzaafaGaeyypa0JaeGimaadabaGaeyOhIukaniabggHiLdaaleaacqWGSbaBcqGH9aqpcqaIWaamaeaacqGHEisPa0GaeyyeIuoaaSqaaiabdAfawbqab0Gaey4kIipakiabdMfaznaaBaaaleaacqWGSbaBcqWGTbqBaeqaaOGaeiikaGIae8hUde3aaSbaaSqaaiabigdaXaqabaGccqGGSaalcqWFgpGzdaWgaaWcbaGaeGymaedabeaakiabcMcaPiabdMfaznaaBaaaleaacuWGSbaBgaqbaiqbd2gaTzaafaaabeaakiabcIcaOiab=H7aXnaaBaaaleaacqaIXaqmaeqaaOGaeiilaWIae8NXdy2aaSbaaSqaaiabigdaXaqabaGccqGGPaqkcqWIAecPdaWgaaWcbaGaemiBaWMaemyBa0gabeaakiabcIcaOiqbdIha4zaalaWaaSbaaSqaaiabigdaXaqabaGccqGGPaqkcqWIAecPdaWgaaWcbaGafmiBaWMbauaacuWGTbqBgaqbaaqabaGccqGGOaakcuWG4baEgaWcamaaBaaaleaacqaIXaqmaeqaaOGaeiykaKIaemiAaGMaeiikaGIafmiEaGNbaSaadaWgaaWcbaGaeGymaedabeaakiabcYcaSiab=r8a0jabcMcaPiaaxMaacaWLjaWaaeWaaeaacqaIXaqmcqaI2aGnaiaawIcacaGLPaaaaaa@B13E@

The next simplification comes when dividing the volume integral into its radial and angular components:

∫ V d 3 x → = ∫ a R | x → | 2 d | x → | ∫ Ω d Ω ∫ Ω d Ω = ∫ 0 2 π d φ ∫ 0 π sin ⁡ ( θ ) d θ MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaafaqaaeGabaaabaWaa8quaeaacqWGKbazaSqaaiabdAfawbqab0Gaey4kIipakmaaCaaaleqabaGaeG4mamdaaOGafmiEaGNbaSaacqGH9aqpdaWdXbqaaiabcYha8jqbdIha4zaalaGaeiiFaW3aaWbaaSqabeaacqaIYaGmaaGccqWGKbazcqGG8baFcuWG4baEgaWcaiabcYha8bWcbaGaemyyaegabaGaemOuaifaniabgUIiYdGcdaWdrbqaaiabdsgaKjabgM6axbWcbaGaeyyQdCfabeqdcqGHRiI8aaGcbaWaa8quaeaacqWGKbazcqGHPoWvaSqaaiabgM6axbqab0Gaey4kIipakiabg2da9maapehabaGaemizaqgaleaacqaIWaamaeaacqaIYaGmiiGacqWFapaCa0Gaey4kIipakiab=z8aMnaapehabaGagi4CamNaeiyAaKMaeiOBa4MaeiikaGIae8hUdeNaeiykaKIaemizaqMae8hUdehaleaacqaIWaamaeaacqWFapaCa0Gaey4kIipaaaaaaa@6D13@

Defining:

q l l ′ ( | x → 1 | , τ ) = ∑ m = − l l ∑ m ′ = − l ′ l ′ 1 2 l + 1 1 2 l ′ + 1 ∫ Ω 1 d Ω 1 Y l m ( θ 1 , φ 1 ) Y l ′ m ′ ( θ 1 , φ 1 ) ℚ l m ( x → 1 ) ℚ l ′ m ′ ( x → 1 ) h ( x → 1 , τ )       ( 17 ) MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacqWGXbqCdaWgaaWcbaGaemiBaWMafmiBaWMbauaaaeqaaOGaeiikaGIaeiiFaWNafmiEaGNbaSaadaWgaaWcbaGaeGymaedabeaakiabcYha8jabcYcaSGGaciab=r8a0jabcMcaPiabg2da9maaqahabaWaaabCaeaadaWcaaqaaiabigdaXaqaaiabikdaYiabdYgaSjabgUcaRiabigdaXaaadaWcaaqaaiabigdaXaqaaiabikdaYiqbdYgaSzaafaGaey4kaSIaeGymaedaaaWcbaGafmyBa0MbauaacqGH9aqpcqGHsislcuWGSbaBgaqbaaqaaiqbdYgaSzaafaaaniabggHiLdaaleaacqWGTbqBcqGH9aqpcqGHsislcqWGSbaBaeaacqWGSbaBa0GaeyyeIuoakmaapefabaGaemizaqgaleaacqGHPoWvdaWgaaadbaGaeGymaedabeaaaSqab0Gaey4kIipakiabgM6axnaaBaaaleaacqaIXaqmaeqaaOGaemywaK1aaSbaaSqaaiabdYgaSjabd2gaTbqabaGccqGGOaakcqWF4oqCdaWgaaWcbaGaeGymaedabeaakiabcYcaSiab=z8aMnaaBaaaleaacqaIXaqmaeqaaOGaeiykaKIaemywaK1aaSbaaSqaaiqbdYgaSzaafaGafmyBa0MbauaaaeqaaOGaeiikaGIae8hUde3aaSbaaSqaaiabigdaXaqabaGccqGGSaalcqWFgpGzdaWgaaWcbaGaeGymaedabeaakiabcMcaPiablQriKoaaBaaaleaacqWGSbaBcqWGTbqBaeqaaOGaeiikaGIafmiEaGNbaSaadaWgaaWcbaGaeGymaedabeaakiabcMcaPiablQriKoaaBaaaleaacuWGSbaBgaqbaiqbd2gaTzaafaaabeaakiabcIcaOiqbdIha4zaalaWaaSbaaSqaaiabigdaXaqabaGccqGGPaqkcqWGObaAcqGGOaakcuWG4baEgaWcamaaBaaaleaacqaIXaqmaeqaaOGaeiilaWIae8hXdqNaeiykaKIaaCzcaiaaxMaadaqadaqaaiabigdaXiabiEda3aGaayjkaiaawMcaaaaa@9848@

it is possible to write:

ν a R ( τ ) = ( 4 π ) 2 ∑ l = 0 ∞ ∑ l ′ = 0 ∞ ∫ a R d | x → 1 | 1 | x → 1 | l + l ′ q l l ′ ( | x → 1 | , τ )       ( 18 ) MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaaiiGacqWF9oGBdaWgaaWcbaGaemyyaeMaemOuaifabeaakiabcIcaOiab=r8a0jabcMcaPiabg2da9iabcIcaOiabisda0iab=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@72E8@

Now, the angular integral in (17) is over a finite region and since the ℚ and h(x→ MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacuWG4baEgaWcaaaa@2E37@₁, τ) are bounded then so is q_ll'(|x→ MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacuWG4baEgaWcaaaa@2E37@₁|, τ). Let:

| q l l ′ ( | x → 1 | , τ ) | < q l l ′ ∗ ( τ ) ; a ≤ | x → 1 | ≤ R       ( 19 ) MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaadaabdaqaaiabdghaXnaaBaaaleaacqWGSbaBcuWGSbaBgaqbaaqabaGcdaqadaqaamaaemaabaGafmiEaGNbaSaadaWgaaWcbaGaeGymaedabeaaaOGaay5bSlaawIa7aiabcYcaSGGaciab=r8a0bGaayjkaiaawMcaaaGaay5bSlaawIa7aiabgYda8iabdghaXnaaDaaaleaacqWGSbaBcuWGSbaBgaqbaaqaaiabgEHiQaaakiabcIcaOiab=r8a0jabcMcaPiabcUda7iabdggaHjabgsMiJoaaemaabaGafmiEaGNbaSaadaWgaaWcbaGaeGymaedabeaaaOGaay5bSlaawIa7aiabgsMiJkabdkfasjaaxMaacaWLjaWaaeWaaeaacqaIXaqmcqaI5aqoaiaawIcacaGLPaaaaaa@598A@

then:

| ν a R ( τ ) | ≤ ( 4 π ) 2 ∑ l = 0 ∞ ∑ l ′ = 0 ∞ q l l ′ ∗ ( τ ) ∫ a R d | x → 1 | 1 | x → 1 | l + l ′ ≤ ( 4 π ) 2 ∑ l = 0 ∞ ∑ l ′ = 0 ∞ q l l ′ ∗ ( τ ) 1 l + l ′ − 1 ( 1 a l + l ′ − 1 − 1 R l + l ′ − 1 )       ( 20 ) MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacqGG8baFiiGacqWF9oGBdaWgaaWcbaGaemyyaeMaemOuaifabeaakiabcIcaOiab=r8a0jabcMcaPiabcYha8jabgsMiJkabcIcaOiabisda0iab=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@A7EB@

Note that since the sources are either dipolar or quadrupolar then:

q_ll' (|x→ MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacuWG4baEgaWcaaaa@2E37@₁|, τ) = 0 unless l = l' = 1 or 2

so that l+l'-l > 1. This means that as R increases without limit the covariance function remains bounded by the quantity:

lim ⁡ x → ∞ | ν a R ( τ ) | < ( 4 π ) 2 ∑ l = 0 ∞ ∑ l ′ = 0 ∞ q l l ′ ∗ ( τ ) 1 l + l ′ − 1 ( 1 a l + l ′ − 1 ) MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaadaWfqaqaaiGbcYgaSjabcMgaPjabc2gaTbWcbaGaemiEaGNaeyOKH4QaeyOhIukabeaakiabcYha8HGaciab=17aUnaaBaaaleaacqWGHbqycqWGsbGuaeqaaOGaeiikaGIae8hXdqNaeiykaKIaeiiFaWNaeyipaWJaeiikaGIaeGinaqJae8hWdaNaeiykaKYaaWbaaSqabeaacqaIYaGmaaGcdaaeWbqaamaaqahabaGaemyCae3aa0baaSqaaiabdYgaSjqbdYgaSzaafaaabaGaey4fIOcaaaqaaiqbdYgaSzaafaGaeyypa0JaeGimaadabaGaeyOhIukaniabggHiLdaaleaacqWGSbaBcqGH9aqpcqaIWaamaeaacqGHEisPa0GaeyyeIuoakiabcIcaOiab=r8a0jabcMcaPmaalaaabaGaeGymaedabaGaemiBaWMaey4kaSIafmiBaWMbauaacqGHsislcqaIXaqmaaWaaeWaaeaadaWcaaqaaiabigdaXaqaaiabdggaHnaaCaaaleqabaGaemiBaWMaey4kaSIafmiBaWMbauaacqGHsislcqaIXaqmaaaaaaGccaGLOaGaayzkaaaaaa@6E57@

which is finite. Thus, contributions to the autocovariance function from neural structures far from the electrode are small when the activity in different regions is not correlated and as expected from the prominent appearance of the radius of the electrode, a, is mainly determined by neural structures near the electrode. This conclusion is not influenced by the orientation or the distribution of multipoles within the volume as long as the multipole density remains finite.

In Appendix A, the signal autocovariance function ν_aR (τ) is computed for a more general covariance function for integer values of p:

g ( x → 1 − x → 1 ′ , x → 1 , τ ) = ∑ p = p min ⁡ p max ⁡ h p ( x → 1 , τ ) | x → 1 − x → 1 ′ | p       ( 21 ) MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacqWGNbWzcqGGOaakcuWG4baEgaWcamaaBaaaleaacqaIXaqmaeqaaOGaeyOeI0IafmiEaGNbaSaadaWgaaWcbaGaeGymaedabeaaiiaakiab=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@6A0F@

Although this mathematical argument is more complex, the conclusion is that quadrupolar (l = 2) sources far from an electrode will not significantly contribute to the signal covariance function as long as p_min > 0 and p_max is finite. For dipolar sources (l = 1) the neuronal correlation function, the autocovariance function in general does not have a defined limit as R → ∞ for correlation functions of the form (21) with p_max < 3 as required for convergence of the integrals required for computation of the autocorrelation function.

It is also possible to estimate the dependence of the cross covariance function on the distance between the recording electrodes. In Appendix C it is demonstrated that, in the case where the activity in various neural structures is uncorrelated (i.e. source correlation function of the type (14)), if the sources are dipolar (l, l' = 1) then νaR(0,s→,τ)(4π)2=1|s→| MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaadaWcaaqaaGGaciab=17aUnaaBaaaleaacqWGHbqycqWGsbGuaeqaaOGaeiikaGIaeGimaaJaeiilaWIafm4CamNbaSaacqGGSaalcqWFepaDcqGGPaqkaeaacqGGOaakcqaI0aancqWFapaCcqGGPaqkdaahaaWcbeqaaiabikdaYaaaaaGccqGH9aqpdaWcaaqaaiabigdaXaqaaiabcYha8jqbdohaZzaalaGaeiiFaWhaaaaa@44DA@. If the sources are quadrupolar, then l, l' = 2 and νaR(0,s→,τ)(4π)2=1|s→|3 MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaadaWcaaqaaGGaciab=17aUnaaBaaaleaacqWGHbqycqWGsbGuaeqaaOGaeiikaGIaeGimaaJaeiilaWIafm4CamNbaSaacqGGSaalcqWFepaDcqGGPaqkaeaacqGGOaakcqaI0aancqWFapaCcqGGPaqkdaahaaWcbeqaaiabikdaYaaaaaGccqGH9aqpdaWcaaqaaiabigdaXaqaaiabcYha8jqbdohaZzaalaGaeiiFaW3aaWbaaSqabeaacqaIZaWmaaaaaaaa@45FB@. The critical observation is that even when the activity in the neural structures is uncorrelated, the measured cross covariance functions decline as power-law function of the distance between the electrodes and do not display the expected delta function behavior.

Furthermore, as shown in Appendix D, when the neuronal correlation function is of the power law form and the neuronal structures produce a predominantly dipolar field, the measured correlation declines more slowly with distance than the neuronal correlation function. In the setting of quadrupolar sources, the measured cross covariance function has the same dependence on the distance between the electrodes as the neuronal correlation function.

The arguments presented above and in Appendix A formally refer to recordings from a monopolar electrode. However, as demonstrated in Appendix B, similar conclusions can be drawn for a complex electrode whose potential is the linear combination of the potentials at a number of other electrodes. In particular, in Appendix B, it is demonstrated that bipolar recording of dipolar sources is equivalent to monopolar recording of quadrupolar sources. Thus, although monopolar recordings from highly correlated dipole sources (as characterized by 0 <p_min ≤ 2 in equation (21)) are dominated by sources distance from the electrode, bipolar recordings from such highly correlated dipole sources receive the largest contribution from sources close to the electrode. Thus, the recordings from different electrodes can be vastly different when they are recording from extended neural structures with long-range correlations.

In order to obtain a qualitative understanding of these mathematical arguments, it is helpful to return to the original argument of section 2.1. From the discussion above, it is clear that that the key element left out of the original argument was the finite range of neuronal correlations. It is instructive to consider a very crude qualitative argument which takes into account these finite range effects. Assume that the neural system can be divided up into discrete elements of volume v₀ and label each element with the index i, then the recorded potential can be written:

V = ℚ 0 ∑ i = 1 N 1 r i m σ i MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacqWGwbGvcqGH9aqpcqWIAecPdaWgaaWcbaGaeGimaadabeaakmaaqahabaWaaSaaaeaacqaIXaqmaeaacqWGYbGCdaqhaaWcbaGaemyAaKgabaGaemyBa0gaaaaaaeaacqWGPbqAcqGH9aqpcqaIXaqmaeaacqWGobGta0GaeyyeIuoaiiGakiab=n8aZnaaBaaaleaacqWGPbqAaeqaaaaa@40DB@

where ℚ₀ is the moment associated with each element, r_i is the distance of the i' th element from the electrode, σ_i reflects that state of activation of the i'th element and N is the total number of elements:

N = 4 3 π R 3 V 0 MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacqWGobGtcqGH9aqpdaWcaaqaamaalaaabaGaeGinaqdabaGaeG4mamdaaGGaciab=b8aWjabdkfasnaaCaaaleqabaGaeG4mamdaaaGcbaacbaGae4Nvay1aaSbaaSqaaiabicdaWaqabaaaaaaa@3750@

where R is the radius of the neural system. In order to estimate the potential, it is reasonable to note that most of the distances, r_i, are on the order of R so that:

V ≈ ℚ 0 R m ∑ i = 1 N σ i MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacqWGwbGvcqGHijYUdaWcaaqaaiablQriKoaaBaaaleaacqaIWaamaeqaaaGcbaGaemOuai1aaWbaaSqabeaacqWGTbqBaaaaaOWaaabCaeaaiiGacqWFdpWCdaWgaaWcbaGaemyAaKgabeaaaeaacqWGPbqAcqGH9aqpcqaIXaqmaeaacqWGobGta0GaeyyeIuoaaaa@3EFB@

The signal variance is given by:

〈 ( V 2 − 〈 V 〉 ) 2 〉 ≈ ( ℚ 0 R m ) 2 〈 ( ∑ i = 1 N [ σ i − 〈 σ i 〉 ] ) 2 〉 MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaadaaadeqaamaabmaabaGaemOvay1aaWbaaSqabeaacqaIYaGmaaGccqGHsisldaaadeqaaiabdAfawbGaayzkJiaawQYiaaGaayjkaiaawMcaamaaCaaaleqabaGaeGOmaidaaaGccaGLPmIaayPkJaGaeyisIS7aaeWaaeaadaWcaaqaaiablQriKoaaBaaaleaacqaIWaamaeqaaaGcbaGaemOuai1aaWbaaSqabeaacqWGTbqBaaaaaaGccaGLOaGaayzkaaWaaWbaaSqabeaacqaIYaGmaaGcdaaadeqaamaabmaabaWaaabCaeaadaWadaqaaGGaciab=n8aZnaaBaaaleaacqWGPbqAaeqaaOGaeyOeI0YaaaWabeaacqWFdpWCdaWgaaWcbaGaemyAaKgabeaaaOGaayzkJiaawQYiaaGaay5waiaaw2faaaWcbaGaemyAaKMaeyypa0JaeGymaedabaGaemOta4eaniabggHiLdaakiaawIcacaGLPaaadaahaaWcbeqaaiabikdaYaaaaOGaayzkJiaawQYiaaaa@57ED@

Defining:

〈 ( σ i − 〈 σ i 〉 ) 2 〉 = σ 0 2 MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaadaaadeqaamaabmaabaacciGae83Wdm3aaSbaaSqaaiabdMgaPbqabaGccqGHsisldaaadeqaaiab=n8aZnaaBaaaleaacqWGPbqAaeqaaaGccaGLPmIaayPkJaaacaGLOaGaayzkaaWaaWbaaSqabeaacqaIYaGmaaaakiaawMYicaGLQmcacqGH9aqpcqWFdpWCdaqhaaWcbaGaeGimaadabaGaeGOmaidaaaaa@3F68@

the term:

〈 ( ∑ i = 1 N [ σ i − 〈 σ i 〉 ] ) 2 〉 ≈ σ 0 2 N N c MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaadaaadeqaamaabmaabaWaaabCaeaadaWadaqaaGGaciab=n8aZnaaBaaaleaacqWGPbqAaeqaaOGaeyOeI0YaaaWabeaacqWFdpWCdaWgaaWcbaGaemyAaKgabeaaaOGaayzkJiaawQYiaaGaay5waiaaw2faaaWcbaGaemyAaKMaeyypa0JaeGymaedabaGaemOta4eaniabggHiLdaakiaawIcacaGLPaaadaahaaWcbeqaaiabikdaYaaaaOGaayzkJiaawQYiaiabgIKi7kab=n8aZnaaDaaaleaacqaIWaamaeaacqaIYaGmaaGccqWGobGtcqWGobGtdaWgaaWcbaGaem4yamgabeaaaaa@4C96@

where N_c is the number of other elements with which the given element is correlated. If the correlation distance is R_C, then:

N c ≈ M i n ( 4 3 π R 3 V 0 , 4 3 π R c 3 V 0 ) MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacqWGobGtdaWgaaWcbaGaem4yamgabeaakiabgIKi7Iqaaiab=1eanjab=LgaPjab=5gaUnaabmaabaWaaSaaaeaadaWcaaqaaiabisda0aqaaiabiodaZaaaiiGacqGFapaCcqWGsbGudaahaaWcbeqaaiabiodaZaaaaOqaaiab=zfawnaaBaaaleaacqaIWaamaeqaaaaakiabcYcaSmaalaaabaWaaSaaaeaacqaI0aanaeaacqaIZaWmaaGae4hWdaNaemOuai1aa0baaSqaaiabdogaJbqaaiabiodaZaaaaOqaaiab=zfawnaaBaaaleaacqaIWaamaeqaaaaaaOGaayjkaiaawMcaaaaa@4987@

since the number of elements correlated with a given element can never be larger than the total number of elements in the system. Thus:

〈 ( V 2 − 〈 V 〉 ) 2 〉 = ℚ 0 2 ( 4 3 π V 0 ) 2 M i n ( R , R c ) 3 R 2 m − 3       ( 22 ) MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaadaaadeqaamaabmaabaGaemOvay1aaWbaaSqabeaacqaIYaGmaaGccqGHsisldaaadeqaaiabdAfawbGaayzkJiaawQYiaaGaayjkaiaawMcaamaaCaaaleqabaGaeGOmaidaaaGccaGLPmIaayPkJaGaeyypa0JaeSOgHq6aa0baaSqaaiabicdaWaqaaiabikdaYaaakmaabmaabaWaaSaaaeaadaWcaaqaaiabisda0aqaaiabiodaZaaaiiGacqWFapaCaeaaieaacqGFwbGvdaWgaaWcbaGaeGimaadabeaaaaaakiaawIcacaGLPaaadaahaaWcbeqaaiabikdaYaaakmaalaaabaGaemyta0KaemyAaKMaemOBa4MaeiikaGIaemOuaiLaeiilaWIaemOuai1aaSbaaSqaaiabdogaJbqabaGccqGGPaqkdaahaaWcbeqaaiabiodaZaaaaOqaaiabdkfasnaaCaaaleqabaGaeGOmaiJaemyBa0MaeyOeI0IaeG4mamdaaaaakiaaxMaacaWLjaWaaeWaaeaacqaIYaGmcqaIYaGmaiaawIcacaGLPaaaaaa@5AC4@

so that:

〈 ( V 2 − 〈 V 〉 ) 2 〉 ∼ 1 R 2 m − 6 ; R < < R c 〈 ( V 2 − 〈 V 〉 ) 2 〉 ∼ 1 R 2 m − 3 ; R > > R c MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=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@5FAF@

This qualitative argument suggests that recordings from dipole sources m = 2 are strongly dependent on the size of the neural system when the size of the system is much smaller than the correlation distance but is stable to changes in R when the system is much larger than the correlation distance. The recorded signal from quadrupolar sources m = 3 is independent of R only for the large systems while those from higher order moments m ≥ 4 are always independent of R.