Spike Timing Amplifies the Effect of Electric Fields on Neurons: Implications for Endogenous Field Effects

Predictions of timing changes

Despite mounting phenomenological evidence that small fields can entrain network activity and have an effect on brain function, to date, there is no experimentally verified mechanistic theory on how this causal interaction may occur. We hypothesized that incremental polarization attributable to an extracellular field will affect the time of threshold crossing; under the assumption of a constant firing threshold, we made a number of predictions that will now be described.

For the case of low-frequency or DC extracellular fields with strength E, which polarize a cell by ΔV_E = cE, where c is the susceptibility of the membrane to an extracellular field, we predicted an advance in spike timing of the following: Note that c refers to the susceptibility effective at the site of spike initiation (i.e., the soma). Subscripts emphasize that ΔV_E results from application of the extracellular field, whereas V̇_I is the result of a concurrent intracellular/synaptic current ramp that induces the action potential spike.

For the case of AC fields, we predicted increased coherence of spiking with the ongoing oscillations. Change in coherence can be quantified as a change in the distribution of firing times occurring at phase ϕ within the oscillatory cycle (with duration T). The distribution of firing phase, p(ϕ), for the present experimental preparation can be expressed as follows: assuming that ϕ is within a depolarizing phase of the membrane oscillation and the cell has not yet fired (Fig. 3B, green shaded region); otherwise, p(ϕ) = 0, because only the time to the first spike was analyzed in the experiment (red shaded region). Equation 3 assumes a uniform initial distribution and no noise (for deviation, see supplemental data, available at www.jneurosci.org as supplemental material). Because the membrane acts as a low-pass filter, a delay between the extracellular field and the induced transmembrane polarization phase is expected; this is expressed in Equation 3 as phase delay, α. The phase distribution of Equation 3 can be characterized by the mean phase, Φ, and the vector strength, r, as shown in Figure 3C. Both should increase monotonically with cE/V̇, with mean phase remaining constrained to the range, α ≤ Φ ≤ α + 90° (supplemental Fig. S1, available at www.jneurosci.org as supplemental material).

This series of predictions were born under the experimental conditions tested.

For both DC and AC fields, the strength of the effect on spike timing scales with where G = c/V̇ can be considered as a timing amplification factor. It increases with the susceptibility of the membrane to extracellular polarization (millivolts of transmembrane polarization at the site of action potential initiation per millivolts per millimeter extracellular field) and decreases with the depolarizing synaptic ramp slope (in millivolts per second). For the case of AC fields, the timing amplification factor is further scaled by field frequency (1/T).

Generalization and quantitative parameterization

Equations 2–4 provide a general theoretical framework for considering the effects of any electric field on spike timing in any neuron with appropriate membrane dynamics (see below). Quantitative application requires knowledge of the intracellular (ramp slope) and extracellular (field amplitude and period) waveforms, which are readily measured. For low-frequency (quasi-static) fields, cell susceptibility can be determined using the method outlined above.

Equation 3 can be readily applied to any periodic field with a complex spatiotemporal profile by considering only the maximum positive instantaneous cE (the phase in the field cycle when the induced transmembrane depolarization is maximal). The susceptibility, c, is specific to a given spatiotemporal waveform (e.g., frequency, nonuniformity). The uniform field susceptibility, for any given temporal waveform, presumably sets a lower susceptibility bound for nonuniform fields. The membrane, acting as a low-pass field filter (Bikson et al., 2004) may introduce a frequency-specific phase delay α > 0°, as well as an attenuated (relative to DC) susceptibility. The DC field susceptibility thus sets an upper limit for AC fields of comparable spatial profile. The phase delay may be estimated assuming the membrane acts as a first-order low-pass filter; thus, a membrane time constant of τ = 30 ms should result in a phase delay of α ≈ 40° for 30 Hz fields, consistent with our experimental findings using timing data (Fig. 3).

Applying timing changes to discern susceptibility

As reinforced above, the polarization susceptibility of neurons is of fundamental importance in assessing subsequent effects on nervous system function. The field-induced polarization is inherently variable between neurons attributable in part to alignment with the field, neuronal geometry, and membrane biophysical properties (Rattay, 1998). Here we developed a method using measured timing changes to calculate the sensitivity of the neuron to polarization by an electric field (Fig. 2B). This technique may yield greater accuracy in measuring polarization sensitivity to small (<5 mV/mm) electric fields than can be achieved by conventional methods. Conventional methods are limited because experimental (line) noise is of magnitude comparable with the polarization induced by small fields, and they require an additional, precisely positioned, isopotential field electrode. Our timing-based method may be readily applied for nonuniform fields (e.g., unipolar, bipolar, deep brain stimulation spatial waveforms) and in vivo. As an additional advantage, our timing method may inherently extract the susceptibility at the spike initiation zone. Conventional polarization measurements used here and in previous studies (Bikson et al., 2004) show average susceptibility (coupling constant) values similar to our method using timing changes.

Membrane dynamics and noise

The theory developed here assumes a simple threshold mechanism equivalent to an integrate-and-fire model neuron with constant firing threshold. The accuracy of the assumption of constant threshold depends on a number of factors, including the channel kinetics of the specific cell type and the nature of the incoming synaptic input; experimental and theoretical work show both dynamic and constant thresholds (Fricker et al., 1999; Azouz and Gray, 2000). The present experiments with DC stimulation controlled for potential slope-dependent thresholds by measuring only changes in timing relative to a control condition with the same ramp slope. The experimental data for the AC stimulation is generally consistent with a constant threshold.

Interestingly, our theory on the scaling of timing effects with ramp slope also explains a number of existing observations on the timing jitter that results from synaptic and membrane noise. If noise induces variability in the membrane potential (ΔV_n), we expect that the resulting spike timing jitter will scale with Δt = ΔV_n/V̇. Indeed, under frozen-noise stimulation, precise spike timing is observed (Bryant and Segundo, 1976), with a timing precision that increases for faster stimuli (Mainen and Sejnowski, 1995). Corresponding simulations with a Hodgkin–Huxley neuron subject to membrane noise also show that increased stimulus magnitude increases timing precision (Schneidman et al., 1998). A similar inverse dependence of spike time precision with stimulus intensity can be observed in cortical pyramidal neurons subject to synaptic noise (Shu et al., 2003). According to our theory, both a reduced time constant and an increased magnitude of the stimulus will increase membrane ramp slope and thus reduce timing jitter attributable to noise.

Functional amplification: network consequences

We found that a 1 mV/mm uniform field induced on average a transmembrane potential change of ∼0.1 mV. Compared with the scale of depolarization necessary to bring a neuron from rest to threshold (∼15 mV), these fields were previously considered insignificant with respect to action potential initiation. Previous action potential threshold studies identified changes attributable to electric fields of <5 mV/mm (Jefferys, 1981). Rather than spike generation, here we demonstrated changes in timing, consistent with the proposed amplification mechanism. The present results provide a potential mechanism for the effects on network spike timing demonstrated previously in vitro with exogenous uniform fields as low as 0.1 mV/mm (Deans et al., 2003; Francis et al., 2003; Fujisawa et al., 2004) and in vivo with calculated fields of 1.2 mV/mm (Marshall et al., 2006).

Electric field-induced changes in spike timing would be particularly relevant for temporal coding during coherent (synchronous) network activity. For example, theta-modulated gamma activity in the hippocampus has been identified as a physiological correlate for a number of phenomena, including spatial navigation, memory, etc. (Miller, 1991; Kahana et al., 2001; Canolty et al., 2006). Extracellular gamma activity amplitudes of up to 0.2 mV are observed in vivo (Csicsvari et al., 2003) and in vitro (Hajos et al., 2004; Mann et al., 2005), with complete phase reversal across 100 μm (from stratum pyramidale to stratum radiatum) resulting in field gradients of 4 mV/mm. An estimated gamma field-induced polarization of at least ΔV_E = 0.4 mV (uniform field sensitivity providing a lower bound for the coupling factor) is small compared with the full-scale voltage fluctuations (e.g., AP threshold); however, considering a theta oscillation synaptic ramp with slope of V̇_I = 0.2 mV/ms (15 mV change between resting and firing potentials within 75 ms of half a theta cycle), a time shift of Δt = 1 ms, and a Rayleigh vector strength of r = 0.39 with a mean angle of α can be estimated. Thus, extracellular fields oscillating at gamma frequency may have a functional effect on AP timing during theta activity even in a single cycle. Moreover, because gamma field activity is coherent across several millimeters of the pyramidal layer (Csicsvari et al., 2003; Mann et al., 2005), extracellular gamma oscillations could facilitate coherence across a large population of neurons. Finally, computational studies have demonstrated that an additional synchronizing effect may result during recurrent network activity when small changes in time add up over multiple cycles (Parra and Bikson, 2004).

The discussion above assumes that the timing of extracellular fields relative to synaptic input acts to promote coherent firing by advancing late spikes and delaying early spikes. However, a different temporal relationship between fields and underlying network activity could also have the opposite effect of advancing early spikes and delaying late spikes, thus acting as a safety mechanism to prevent hypersynchronization.

Our approach for predicting field-induced timing changes (Eqs. 2, 3) can be readily extended to other networks using readily measurable extracellular and intracellular waveforms and using either field susceptibility estimates or our novel method to measure susceptibility. Indeed, extracellular field oscillations are a ubiquitous marker for network oscillations (Chrobak and Buzsaki, 1998; Donoghue et al., 1998, Csicsvari et al., 2003; Nunez and Srinivasan, 2005; Sarnthein and Jeanmonod, 2007). More than an epiphenomenon, our results indicate a functional role for field potentials in modulating spike timing. Indeed, the information content of endogenous extracellular potential oscillations is often comparable with multiunit activity (Scherberger et al., 2005; Heldman et al., 2006; Liu and Newsome, 2006). Synaptic mechanisms underlie the generation of many classes of oscillations (Bragin et al., 1995; McBain et al., 1999; Buzsaki et al., 2003; Jones and Wilson, 2005; Netoff et al., 2005; Siapas et al., 2005); field-effect coupling is not proposed as alternative but rather as a complementary mechanism with unique spatiotemporal features.

Our system does not presuppose the source, exogenous or endogenous, of the modulating electric fields. Thus, the equations developed in this paper can be similarly applied in predicting the effects of environmental electric fields; electromagnetic field safety standards have not previously considered effects of timing. Finally, our results support the development of therapeutic stimulation technologies targeting neuronal timing; abnormal timing is indeed a hallmark of many neurological disorders, and subthreshold stimulation approaches may be readily adapted for non-invasive (transcranial) technologies.