Extending Integrate-and-Fire Model Neurons to Account for the Effects of Weak Electric Fields and Input Filtering Mediated by the Dendrite

Abstract

Transcranial brain stimulation and evidence of ephaptic coupling have recently sparked strong interests in understanding the effects of weak electric fields on the dynamics of brain networks and of coupled populations of neurons. The collective dynamics of large neuronal populations can be efficiently studied using single-compartment (point) model neurons of the integrate-and-fire (IF) type as their elements. These models, however, lack the dendritic morphology required to biophysically describe the effect of an extracellular electric field on the neuronal membrane voltage. Here, we extend the IF point neuron models to accurately reflect morphology dependent electric field effects extracted from a canonical spatial “ball-and-stick” (BS) neuron model. Even in the absence of an extracellular field, neuronal morphology by itself strongly affects the cellular response properties. We, therefore, derive additional components for leaky and nonlinear IF neuron models to reproduce the subthreshold voltage and spiking dynamics of the BS model exposed to both fluctuating somatic and dendritic inputs and an extracellular electric field. We show that an oscillatory electric field causes spike rate resonance, or equivalently, pronounced spike to field coherence. Its resonance frequency depends on the location of the synaptic background inputs. For somatic inputs the resonance appears in the beta and gamma frequency range, whereas for distal dendritic inputs it is shifted to even higher frequencies. Irrespective of an external electric field, the presence of a dendritic cable attenuates the subthreshold response at the soma to slowly-varying somatic inputs while implementing a low-pass filter for distal dendritic inputs. Our point neuron model extension is straightforward to implement and is computationally much more efficient compared to the original BS model. It is well suited for studying the dynamics of large populations of neurons with heterogeneous dendritic morphology with (and without) the influence of weak external electric fields.

Network remodeling induced by transcranial brain stimulation: A computational model of tDCS-triggered cell assembly formation

“…Most notably, Bikson et al (2006) has explored several aspects of this: extracellular potassium concentration, polarization of the axonal terminal, action potential timing, and inhibitory neurons. Joucla and Yvert (2009) has provided an estimate of membrane potential changes for large axons exposed to an electric field, and Aspart et al (2016) conceived the influence of the electric field on neuronal dendrites as external input to the soma. Another computational approach based on modern neural imaging methods has shed light on the question how strong the stimulation effects actually are.…”

Weak electric fields promote resonance in neuronal spiking activity: analytical results from two-compartment cell and network models

“…Although the electric fields caused by this type of noninvasive intervention exhibit low magnitudes (≤ 1-2 V/m [5,6]) they can modulate neuronal spiking activity [2,12,13] and lead to changes in cognitive processing [14,15], offering a number of possible clinical interventions [16,17]. The influence of extracellular fields on the subthreshold membrane voltage of single cells has been thoroughly studied and biophysically explained [12,[18][19][20][21]. How weak electric fields affect neuronal spiking activity and interact with network dynamics, however, is currently not well understood.…”

“…Multi-compartment models are useful tools to dissect effects in single neurons in the absence of input fluctuations [22][23][24], but they are not well suited to study neuronal and network spiking activity in noisy, in-vivo like conditions, because of their large complexity. Single-compartment (point) neuron models on the other hand allow for effective mechanistic analyses at the network level (see, e.g., [25][26][27]); however, they lack the spatial structure that is required to biophysically describe the effects of an electric field [20]. Nevertheless, networks of point model neurons have been repeatedly used in conjunction with rough phenomenological implementations of extracellular field effects [2,9,11,28].…”

Low-rate firing limit for neurons with axon, soma and dendrites driven by spatially distributed stochastic synapses

“…Due partly to the sparsity of the experimental data required for model constraint, 36 but also because of the mathematical complexity involved, few analytical results 37 regarding stochastic synaptic integration are available for neurons with dendritic 38 structure, excepting the work of Tuckwell [30][31][32]. Nevertheless there is increasing 39 interest in the integrative and firing response of spatial neuron models [33,34], for 40 neurons subject to and generating electric fields [35][36][37] or the effect of axonal load and 41 position of the action-potential initiation region [34,[38][39][40][41][42]. Advances in optogenetics 42 and multiple, parallel intracellular recordings have made experimental measurement and 43 stimulation of in vivo-like input at arbitrary dendritic locations feasible [43][44][45].…”

“…Dendritic structure has been previously shown to influence the firing rate for deterministic input [64,65]. However, apart from the work of Tuckwell [30][31][32], 386 analytical studies of stochastic drive in extended neuron models have largely focussed 387 on a single dendrite with drive typically applied at a single point [36,39] rather than 388 distributed over the dendrite, or as a two-compartmental model [66]. This study 389 demonstrates that in the low-rate regime, the upcrossing approximation allows for the 390 analytical study of spatial models that need not be limited to a single dendrite nor with 391 stochastic synaptic drive confined to a single point, but distributed as is the case in vivo.…”