Systematic investigations through several experimental techniques have revealed that hippocampal pyramidal neurons express voltage gated ion channels (VGICs) with well-defined gradients along their dendritic arbor. These actively maintained gradients in various dendritic VGICs effectuate several stereotypic, topographically continuous functional gradients along the topograph of the dendritic arbor, and have been referred to as intraneuronal functional maps. The prime goal of my thesis was to understand the emergence and homeostasis of the several coexistent functional maps that express within hippocampal pyramidal neurons. In the first part of the thesis, we focus only on spatial interactions between ion channels and analyzed the role of such interactions in the emergence of functional maps. We developed a generalized quantitative framework, the influence field, to analyze the extent of influence of a spatially localized VGIC cluster. Employing this framework, we showed that a localized VGIC cluster could have spatially widespread influence, and was heavily reliant on the specific physiological property and background conductances. Using the influence field model, we reconstructed functional gradients from VGIC conductance gradients, and demonstrated that the cumulative contribution of VGIC conductances in adjacent compartments plays a critical role in determining physiological properties at a given location. These results suggested that spatial interactions among spatially segregated VGIC clusters are necessary for the emergence of the functional maps. In the second part of the thesis, we assessed the specific roles of only kinetic interactions between ion channels in determining physiological properties by employing a single-compartmental model. In doing this, we analyzed the roles of interactions among several VGICs in regulating intrinsic response dynamics. Using global sensitivity analysis, we showed that functionally similar models could be achieved even when underlying parameters displayed tremendous variability and exhibited weak pair-wise correlations. These results suggested that that response homeostasis could be achieved through several non-unique channel combinations, as an emergent consequence of kinetic interactions among these channel conductances. In the final part of the thesis, we analyzed the combined impact of both spatial and kinetic interactions among ion channel conductances on the emergence and homeostasis of functional maps in a neuronal model endowed with extensive dendritic arborization. To do this, we performed global sensitivity analysis on morphologically realistic conductance-based models of hippocampal pyramidal neurons that coexpressed six functional maps. We found topographically continuous functional maps to emerge from disparate model parameters with weak pair-wise correlations between parameters. These results implied that individual channel properties need not be set at constant values in achieving overall homeostasis of several coexistent functional maps. We suggest collective channelostasis, where several channels regulate their properties and expression profiles in an uncorrelated manner, as an alternative for accomplishing functional map homeostasis. Finally, we developed a methodology to assess the contribution of individual channel conductances to the various functional measurements employing virtual knockout simulations. We found that the deletion of individual channels resulted in variable, measurement-and location-specific impacts across the model population.

Local field potentials (LFPs), extracellular potentials that reflect localized electrical activity, have long been used as a window to understand the behavioural dependence and mechanistic aspects of brain physiology. A principal premise that has driven the interpretation of LFPs is that they largely reflect the synaptic drive that impinges on neurons located in the vicinity of the recording microelectrode. An implicit, yet critical, assumption that led to the emergence of this premise is that dendrites, the structures onto which most synaptic inputs project, are purely passive compartments. However, there is a growing body of evidence demonstrating that dendrites express a plethora of active conductance, like voltage-gated ion channels, several of which are active in the subthreshold regime. These subthreshold-activated ion channels and their intra-neuronal localization profiles play widely acknowledged regulatory roles in the physiology, plasticity and pathophysiology of synapses and neurons. Despite this, the implications for the existence of these subthreshold conductances on constituent oscillatory patterns in LFPs and on the phase of neuronal spiking with reference to oscillating LFPs have surprisingly remained unexplored. The aim of this thesis is to examine if there exists a role of subthreshold conductances in regulating LFPs and the phase of spikes with reference to these LFPs. To address this, we chose to study LFPs and spikes from the CA1 region of the rat hippocampus, with hyperpolarization-activated cyclic-nucleotide-gated (HCN) channels forming the specific subthreshold conductance of focus. The reasons behind these choices were manifold. First, CA1 pyramidal neurons are arranged in a laminar open-field configuration, making the interpretation of the source-sink formation in this region relatively tractable. Second, the dendrites of these neurons are endowed with a multitude of subthreshold conductances whose expression profiles, physiology and plasticity have been characterized in great detail. Third, this brain region has been implicated in coding for episodic and spatial memories. The phase of the spikes of the CA1 pyramidal neurons, with reference to the LFP, is believed to serve as a code that can be used to decode the location of the animal. Given that the most dominant LFP pattern seen in the CA1 region during such active exploration (and possibly encoding of spatial memories) consists of oscillations in the 4–10 Hz theta frequency band, we decided to focus our study on theta-frequency LFPs. Finally, consistent with the choice of the specific band of LFP frequencies, we focused on HCN channels because of their predominantly dendritic expression and their ability to bestow resonance and impedance phase lead, both in the theta-frequency range, on CA1 pyramidal neurons. In exploring the role of HCN channels on LFPs, we used a multi-compartmental morphologically realistic CA1 pyramidal neuron model and introduced an HCN channel conductance gradient that was constrained with several experimental measurements. This neuron was driven by dendritic excitatory synapses and perisomatic inhibitory synapses, both theta-modulated with a phase difference of +60º between their arrivals timings. We increased the excitatory synaptic conductance with distance from the soma to account for the fact that irrespective of the location of the synapse in the dendrites, the unitary excitatory post-synaptic potential remains the same at the soma. Employing these model configurations, we generated 25 different synaptic distributions on the same neuronal morphology to account for the input variability and for each of these models, we recorded transmembrane currents from all the compartments, for 8–10 cycles of the theta-modulated inputs. To model LFPs using the forward modelling scheme of line source approximation, we designed a cylindrical neuropil of 40 µm height and 100 µm radius and inserted a virtual linear electrode with 7 contact points distributed on the probe at the canter of the neuropil such that we could compute the LFP at all the strata of the CA1 region. Accounting for the volume of the neuropil and the density of neurons in this region, we took 440 instances of the morphology, rotated them at uniformly distributed angles, and distributed the somata of these model neurons within the neuropil. Each of these 440 neurons received transmembrane currents from one of the 25 models picked uniformly. With a passive model, where we did not introduce HCN channels, we expectedly observed the formation of a source-sink structure that expressed as a progressive phase shift spanning different strata, owing to the perisomatic inhibitory currents coupled with the dendritic excitatory currents. On introducing a somatodendritic gradient of HCN conductance with identical input conditions, we observed a phase lead in the LFPs across all the layers, with the magnitude of the lead increasing with distance from the soma in a manner that was correlated with the increase in HCN conductance. Next, we computed spike phases, for each of the 25 neuron models, with reference to the stratum pyramidale (SP) LFP for model configurations with and without HCN channels. We found that the spikes showed a phase lag in the presence of a gradient of HCN channels when compared to the spike phases measured from the passive neuron models. Finally, we computed the coherence of spikes across all the 25 passive or 25 active (with HCN channels) neuron models and found that the presence of HCN channels greatly enhanced spike phase coherence across neurons. Together, these results demonstrate that the presence of HCN channels introduces a lead in the theta-frequency LFP phase, a lag in the associated spike phase, and a significant enhancement of spike phase coherence. Exploring the robustness of these findings to the model configuration, we first found these conclusions to be robust to increases in neuropil size (400-µm diameter neuropil with 1797 neurons, and 1-mm diameter neuropil with 11297 neurons). Next, we introduced heterogeneities in the population of neurons (in terms of morphology as well as passive and active properties) that formed the neuropil, and found our conclusions to be invariant to such degeneracy in the underlying neuronal population. It has been observed that under certain pathological conditions like epilepsy, an entire population of CA1 neurons can undergo intrinsic plasticity, such as global (i.e., across the entire neuronal topograph) downregulation of HCN channels. To assess the impact of such up/downregulation on LFPs, we respectively increased/decreased HCN channel conductance globally in our model neurons, and found the magnitude of the lead in the LFP phase to progressively increase with HCN-channel conductance. Similarly, the magnitude of the spike-phase lag and the spike phase coherence also progressively increased as functions of HCN-channel conductance. Although such population-level global intrinsic plasticity is observed under pathological conditions, a more physiological scenario would be when a single neuron, in the process of encoding new inputs (such as encoding spatial or episodic memories), undergoes intrinsic plasticity. To assess this, we increased or decreased HCN-channel conductance specifically in a single neuron placed closest to the electrode, while leaving the HCN expression in other neurons of the neuropil at the baseline level. Expectedly, we did not find significant changes in LFP amplitude or phase, but we did find a significant lag in the spike phase preference of the neuron that underwent an upregulation of HCN conductance. Another physiological scenario is when the rat experiences a reward or exhibits anxiety-like behaviour, which can lead to changes in hormonal or neuromodulator concentrations. These changes, functioning through the activation of G-protein coupled receptors and the consequent elevation of cytosolic cyclic adenosine monophosphate (cAMP) concentrations, could shift the half-maximal activation voltage ( V1/2 ) of HCN channels to a more depolarized potential. Would such a shift in V1/2 impact LFPs and spike phases in a manner similar to that observed with increasing the conductance of HCN channels? Assessing this within our modeling framework, we found that shifting the V1/2 by +5 mV resulted in an increased lead in the LFP phase, an increased lag in the spike phase and an enhanced spike phase coherence compared to the case with a hyperpolarized V1/2 . What are the biophysical mechanisms that underlie these robust changes observed in LFPs and spike phases observed as a consequence of these several ways of increasing the current through HCN channels? We reasoned that our observations could be explained by one of the two distinct changes conferred on CA1 pyramidal neuron physiology by the presence of HCN channels. First, in the presence of HCN channels, the voltage response of CA1 pyramidal neurons shows a phase lead with reference to a sinusoidal current input (inductive phase lead) in the theta frequency range. Second, HCN channels regulate the excitability of these cells by decreasing the input resistance and impedance amplitude. To delineate the differential role of the inductive changes vs. changes in excitability, we replaced HCN channels by a faster variant (HCNFast) such that neuronal excitability remained the same while abolishing the inductive phase lead in the theta band. On doing so, we found that the lead in the LFP phase and the lag in the spike phase brought about by HCN channels was partially reversed when HCN conductance values were low. However the reversal was not substantial when HCN conductance values were high, suggesting that the inductive phase component dominates at lower HCN channel conductances, whereas the excitability component plays a critical role at higher HCN conductances. Akin to intrinsic plasticity mentioned above, under certain pathological conditions, an entire population of neurons can undergo scaling of their excitatory or inhibitory synapses. In assessing the implications for such synaptic plasticity, we first found that our conclusions on the roles of HCN channels in introducing a lead in the LFP phase, a lag in the spike phase and an enhancement of spike phase coherence were invariant to the specific values of synaptic conductances, or the phase difference between excitatory and inhibitory theta-modulated inputs. While these observations further established the robustness of the changes brought about by HCN channels to LFPs and associated spikes, we next asked whether synaptic plasticity, mediated by changes in subthreshold synaptic conductances, could itself bring about changes in the LFP and spike phase. Expectedly, we found that scaling up of excitatory synapses introduced a mild lag in the LFP phase and a lead in the spike phase, whereas scaling up of inhibitory synapses introduced a lead in the LFP phase and a lag in the spike phase. Finally, we observed a critical role of the arrival phase of inhibition with reference to excitation in altering both, the stratum pyramidale LFP and associated spike phases, with the magnitude of change in both the LFP and the spike phase roughly following the magnitude of the shift in the excitatory-inhibitory phase difference. However, in contrast to changes observed with HCN-channel plasticity, there was no significant change in spike phase coherence with any of the three forms of synaptic changes explored. Together, our results identify definite roles for HCN channels and synaptic receptors in phase-coding schemas and in the formation and dynamic reconfiguration of neuronal cell assemblies and present a clear case for the incorporation of subthreshold-activated ion channels, their gradients, and their plasticity into the computation of LFPs. Given the rich expression of several subthreshold ion channels — including HCN, A-type potassium and T-type calcium — in neuronal dendrites, future work could focus on the impact of subthreshold channels on LFPs recorded in different brain regions under different behavioral states. This thesis is organized into seven chapters. Chapter 1 provides the motivations for the study, introduces the aim of the study and poses the specific questions asked in our endeavor to understand the role of subthreshold conductances in regulating LFPs and spike phases. Chapter 2 discusses the physiological foundations and relevant literature that places the questions posed in the first chapter in the context of the aim of the thesis, with an emphasis on the literature on HCN channels. In chapter 3, we introduce the computational and theoretical foundations required to model neurons and to compute LFPs. In chapter 4, we look at the consequences of the presence of a non-uniform density of somatodendritic HCN channels on LFPs and spike phase and test the robustness of the effects observed. In chapter 5, we present our assessment of the impact of intrinsic plasticity/modulation of HCN channels on LFPs and spike phases, also exploring the biophysical mechanisms underlying such an impact. In chapter 6, we test if the observed effects still hold under synaptic plasticity, and assess the regulation of LFPs and spike phases by synaptic changes. In chapter 7, we summarize and conclude the results presented in the preceding chapters and provide some potential directions for future studies.