The current level of understanding of how extracellular eld interactions take place, endogenously between neurons and between articially imposed elds and neuronal networks, is still being rened. In particular, there are two aspects which current modeling studies miss and which we want to address. Firstly, often a one-way eld coupling is taken into account, where either the extracellular eld inuences the neuron but is not in turn inuenced by the neuronal activity [56, 57] or the neuron inuences the extracellular eld but is not in turn inuenced by the extracellular activity [58]. Secondly, the three-dimensional structure of the neuronal geometry is
(a) 100−1 101 103 105 107 109 0.5 1 F requency(Hz) N or m a li z ed a m p li tu d e ( m V ) Current injected F ield stimulated (b)
Figure 1.10: Two alternative ways of cell stimulation. (a) (Left) Current injection via whole cell patch-clamp: The membrane behaves isopotentially and the cell response is dependent on the membrane conductance and capacitance, not on the extra and intracellular conductivities (Eq. 1.24). (Right) Membrane potential distribution on a spherical neuron of radius 10µm in the presence of an electric eld oriented along
thex-axis. The membrane potential is dependent on the position on the cell as well
as the extra and intracellular conductivities (Eq. 1.21). (b) Normalized membrane potential responses of a passive current-injected and a passive eld-stimulated spher- ical neuron (of radius10 µm) to sinusoidal stimulation at varying frequencies. The
current-injected point-neuron response (Eq. 1.24) falls o at around 10 Hz stimula- tion frequency, whereas the eld-stimulated response (Eq. 1.21) is sustained up to kHz frequencies. Parameter values are as in table 1.1.
not always fully taken into account as in the example of cable theory [19,56,57]. Our aim is to build three-dimensional, two-way coupled models which fully incorporate the feedback, which in reality exists between the neuron and the extracellular eld as demonstrated recently by Frohlich and McCormick [59]. This will provide for realistic modeling of eld eects on the nervous system in pathological, physiological and clinical contexts. The models will require a synthesis of two dierent modeling approaches used in two dierent elds; rst the nonlinear time-varying behaviour of the membrane characterized by the Hodgkin Huxley equations [18] used exten- sively in theoretical and computational neuroscience, secondly the volume conductor behaviour in the bulk media found by the solution of Maxwell's equations.
In chapter 2 we explore the spatial dependence highlighted above (section 1.6), comparing the cable theory, nite-dierence and nite-element methodologies. We expand the theoretical analysis above [4654] to geometrically extended neurons
ship between the neuron shape and its response at varying eld frequencies. We use the nite-dierence and nite-element methods for solving the analytically awkward problem of a nite cylinder in an oscillating electric eld. We compare these results with those obtained through the extracellular cable equation and delineate the re- lationship between cellular shape, orientation and susceptibility to high-frequency electric elds. In particular we nd that the neurons stimulated by extracellular elds exhibit an eective membrane time constant dependent on their electrotonic length and thus resolve the discrepancy in the literature, highlighted in section 1.6. In the context of cable theory (see section 2.2), the electrotonic length, Le is the physical length of the neurite under consideration (L), measured in units of the
electrotonic length constant
λ=
r
a
2rLgL (1.26)
where a is the radius of the cable, rL is the intracellular resistivity and gL is the membrane leak conductance per unit area as before. λrepresents the distance over
which a current input in to the dendrite will spread and hence inuence the mem- brane potential. Physically, a larger a and a smaller rL would impose a smaller resistance to current spread (leading to a largerλ), whereas a larger gL would lead to more current escaping in to the extracellular space (and hence a smallerλ).
In chapter 3 we investigate the behaviour of the passive cylindrical cell un- der point-source stimulation. Deriving the cable results using Green's functions, we compare them against the results obtained through our nite-element model. Due to the breaking of the axial symmetry by the non-uniform point-source eld, a three-dimensional nite-element model is constructed. Both modeling methodologies reveal a novel form of localized frequency preference by the entirely passive neurons, the magnitude and frequency of which is dependent on the cell geometry and the distance between the neuron and the point-source. To our knowledge this passive resonance has not been reported in the literature before.
of a quasi-active membrane into our eld-neuron system known to lead to reso- nance at a characteristic stimulation frequency, manifesting itself as a peak in the induced voltage oscillations, when stimulated by current injection via whole-cell patch clamp [60,61]. This phenomenon, found in many parts of the central nervous system, including in neocortical neurons [6265] as well as hippocampal pyramidal cells [66,67] and interneurons [67], arises from an interplay between the passive and active neuronal properties. In particular, resonance requires voltage-gated currents, which slowly oppose membrane potential changes [60]. Here we apply the linearized quasi-active membrane model to the spherical and the cylindrical eld-neuron sys- tems, elucidating the relationship between the neuronal shape and its subthreshold resonance in the presence of an oscillating electric eld.
We then go on to construct a fully active eld-neuron nite-element model incorporating Hodgkin Huxley type channels into the membrane. A signicant and novel step towards modeling eld-neuron systems with full two-way feedback between the neuron and the eld, we use the spiking model to demonstrate that the passive resonance and high-frequency response of the passive neurons under point-source stimulation translates to the fully active neuron case.
Lastly, we go further and simulate the frequency response of passive neurons embedded in semi-innite arrays and exposed to extracellular elds, validating the results obtained for the case of isolated neurons.
The work in this thesis can be categorized both as technique development and as nding novel aspects of eld-neuron interaction. The ndings below in relation to the frequency response of neurons to oscillating electric elds have important im- plications for understanding how endogenous and articial electric elds may aect the behaviour of neuronal networks, a eld which has been relatively understudied. Although the primary aim of the project was not technique development, the nite- element models developed for eld-neuron interactions here, could have a greater impact on the eld. Our FEM models have given us the capability to quickly and
eciently model neurons with realistic geometries and biophysics, and with full eld coupling in a tissue environment. Given sucient computational power, this ad- vance could help answer the long standing questions about endogenous eld-neuron interactions and also how articial eld intervention aects the nervous system.