Branching dendrites with resonant membrane: a “sum-over-trips” approach

Dendrites form the major components of neurons. They are complex branching structures that receive and process thousands of synaptic inputs from other neurons. It is well known that dendritic morphology plays an important role in the function of dendrites. Another important contribution to the response characteristics of a single neuron comes from the intrinsic resonant properties of dendritic membrane. In this paper we combine the effects of dendritic branching and resonant membrane dynamics by generalising the “sum-over-trips” approach (Abbott et al. in Biol Cybernetics 66, 49–60 1991). To illustrate how this formalism can shed light on the role of architecture and resonances in determining neuronal output we consider dual recording and reconstruction data from a rat CA1 hippocampal pyramidal cell. Specifically we explore the way in which an Ih current contributes to a voltage overshoot at the soma

Travelling waves in a model of quasi-active dendrites with active spines

Dendrites, the major components of neurons, have many different types of branching structures and are involved in receiving and integrating thousands of synaptic inputs from other neurons. Dendritic spines with excitable channels can be present in large densities on the dendrites of many cells. The recently proposed Spike-Diffuse-Spike (SDS) model that is described by a system of point hot-spots (with an integrate-and-fire process) embedded throughout a passive tree has been shown to provide a reasonable caricature of a dendritic tree with supra-threshold dynamics. Interestingly, real dendrites equipped with voltage-gated ion channels can exhibit not only supra-threshold responses, but also sub-threshold dynamics. This sub-threshold resonant-like oscillatory behaviour has already been shown to be adequately described by a quasi-active membrane. In this paper we introduce a mathematical model of a branched dendritic tree based upon a generalisation of the SDS model where the active spines are assumed to be distributed along a quasi-active dendritic structure. We demonstrate how solitary and periodic travelling wave solutions can be constructed for both continuous and discrete spine distributions. In both cases the speed of such waves is calculated as a function of system parameters. We also illustrate that the model can be naturally generalised to an arbitrary branched dendritic geometry whilst remaining computationally simple. The spatio-temporal patterns of neuronal activity are shown to be significantly influenced by the properties of the quasi-active membrane. Active (sub- and supra-threshold) properties of dendrites are known to vary considerably among cell types and animal species, and this theoretical framework can be used in studying the combined role of complex dendritic morphologies and active conductances in rich neuronal dynamics

Data-driven reduction of dendritic morphologies with preserved dendro-somatic responses

Dendrites shape information flow in neurons. Yet, there is little consensus on the level of spatial complexity at which they operate. Through carefully chosen parameter fits, solvable in the least-squares sense, we obtain accurate reduced compartmental models at any level of complexity. We show that (back-propagating) action potentials, Ca2+ spikes, and N-methyl-D-aspartate spikes can all be reproduced with few compartments. We also investigate whether afferent spatial connectivity motifs admit simplification by ablating targeted branches and grouping affected synapses onto the next proximal dendrite. We find that voltage in the remaining branches is reproduced if temporal conductance fluctuations stay below a limit that depends on the average difference in input resistance between the ablated branches and the next proximal dendrite. Furthermore, our methodology fits reduced models directly from experimental data, without requiring morphological reconstructions. We provide software that automatizes the simplification, eliminating a common hurdle toward including dendritic computations in network models

Dynamics of spatially extended dendrites

Dendrites are the most visually striking parts of neurons. Even so many neuron models are of point type and have no representation of space. In this thesis we will look at a range of neuronal models with the common property that we always include spatially extended dendrites. First we generalise Abbott’s “sum-over-trips” framework to include resonant currents. We also look at piece-wise linear (PWL) models and extend them to incorporate spatial structure in the form of dendrites. We look at the analytical construction of orbits for PWL models. By using both analytical and numerical Lyapunov exponent methods we explore phase space and in particular we look at mode-locked solutions. We will then construct the phase response curve (PRC) for a PWL system with compartmentally modelled dendrites. This sets us up so we can look at the effect of multiple PWL systems that are weakly coupled through gap junctions. We also attach a continuous dendrite to a PWL soma and investigate how the position of the gap junction influences network properties. After this we will present a short overview of neuronal plasticity with a special focus on the spatial effects. We also discuss attenuation of distal synaptic input and how this can be countered by dendritic democracy as this will become an integral part of our learning mechanisms. We will examine a number of different learning approaches including the tempotron and spike-time dependent plasticity. Here we will consider Poisson’s equation around a neural membrane. The membrane we focus on has Hodgkin-Huxley dynamics so we can study action potential propagation on the membrane. We present the Green’s function for the case of a one-dimensional membrane in a two-dimensional space. This will allow us to examine the action potential initiation and propagation in a multi-dimensional axon

Recovery of neuronal channel densities from calcium fluorescence

Neurons have the ability to dynamically adjust their own membrane channel densities to modulate the strength of communication with other neurons. This process is integral to such neuronal functions as spatial recognition and memory but has been difficult to measure experimentally. Historically, neuroscientists have used changes in voltage to infer changes in neuronal channel densities. However, voltage is difficult to measure away from the soma. Many important functions in the neuron, like synaptic integration, take place in the dendritic tree where traditional voltage measurements can not be taken. To interrogate the neuron in the dendrites, experimentalists have come to rely on calcium fluorescence based microscopy to infer qualitative information about voltage changes in the dendrites. In these experiments, intracellular calcium changes due to voltage depolarizations are recorded at spatially distributed sites on the dendrites through the binding of calcium to a fluorescent buffer. The recovery of channel densities can be posed as a parameter identification problem in a coupled nonlinear partial differential equation that relates the responses of calcium, the fluorescent buffer and voltage to neuronal stimulation. We convert temporally and spatially distributed fluorescence data into quantitative measurements of voltage sensitive channel densities by inverting slow time-scaled calcium data into fast time-scaled voltage data. Our approach is to solve four interrelated inverse problems corresponding to three different proposed experiments to go from calcium fluorescence to channel densities. In the first experiment, we use subthreshold calcium dynamics to infer the reaction kinetics between calcium arid fluorescent buffer. From these kinetics, we can use suprathreshold voltage stimulation to infer calcium channel densities and recover distributed voltage data. Finally we use the voltage data to infer potassium channel densities in the dendrites. Our algorithm has been shown to recover channel densities for several different calcium channel models and the delayed rectifying potassium channel from simulated noisy fluorescence data in morphologically realistic neurons

Dynamics of spatially extended dendrites

Dendrites are the most visually striking parts of neurons. Even so many neuron models are of point type and have no representation of space. In this thesis we will look at a range of neuronal models with the common property that we always include spatially extended dendrites. First we generalise Abbott’s “sum-over-trips” framework to include resonant currents. We also look at piece-wise linear (PWL) models and extend them to incorporate spatial structure in the form of dendrites. We look at the analytical construction of orbits for PWL models. By using both analytical and numerical Lyapunov exponent methods we explore phase space and in particular we look at mode-locked solutions. We will then construct the phase response curve (PRC) for a PWL system with compartmentally modelled dendrites. This sets us up so we can look at the effect of multiple PWL systems that are weakly coupled through gap junctions. We also attach a continuous dendrite to a PWL soma and investigate how the position of the gap junction influences network properties. After this we will present a short overview of neuronal plasticity with a special focus on the spatial effects. We also discuss attenuation of distal synaptic input and how this can be countered by dendritic democracy as this will become an integral part of our learning mechanisms. We will examine a number of different learning approaches including the tempotron and spike-time dependent plasticity. Here we will consider Poisson’s equation around a neural membrane. The membrane we focus on has Hodgkin-Huxley dynamics so we can study action potential propagation on the membrane. We present the Green’s function for the case of a one-dimensional membrane in a two-dimensional space. This will allow us to examine the action potential initiation and propagation in a multi-dimensional axon

The spatiotemporal organization of cerebellar network activity resolved by two-photon imaging of multiple single neurons

In order to investigate the spatiotemporal organization of neuronal activity in local microcircuits, techniques allowing the simultaneous recording from multiple single neurons are required. To this end, we implemented an advanced spatial-light modulator two-photon microscope (SLM-2PM). A critical issue for cerebellar theory is the organization of granular layer activity in the cerebellum, which has been predicted by single-cell recordings and computational models. With SLM-2PM, calcium signals could be recorded from different network elements in acute cerebellar slices including granule cells (GrCs), Purkinje cells (PCs) and molecular layer interneurons. By combining WCRs with SLM-2PM, the spike/calcium relationship in GrCs and PCs could be extrapolated toward the detection of single spikes. The SLM-2PM technique made it possible to monitor activity of over tens to hundreds neurons simultaneously. GrC activity depended on the number of spikes in the input mossy fiber bursts. PC and molecular layer interneuron activity paralleled that in the underlying GrC population revealing the spread of activity through the cerebellar cortical network. Moreover, circuit activity was increased by the GABA-A receptor blocker, gabazine, and reduced by the AMPA and NMDA receptor blockers, NBQX and APV. The SLM-2PM analysis of spatiotemporal patterns lent experimental support to the time-window and center-surround organizing principles of the granular layer