Effects of passive dendritic tree properties on the firing dynamics of a leaky-integrate-and-fire neuron
Introduction
Neurons can have extensive spatial geometries, but are typically modeled as single compartment objects, with no explicit spatial dependence. This modeling choice is often made for mathematical tractability and/or computational efficiency. However, many neurons are not electrotonically compact, and single compartment models cannot be expected to capture the full range of dynamical behaviors of neurons. Dendrites (the branched structures emanating from the cell body, or soma) can substantially affect the electrical activity of single neurons. For instance, the types of ion channels and their density along the dendritic tree can alter the firing patterns of neurons [1], [2], [3]. Even dendrites endowed with passive (linear) ion channels can alter the frequency and firing dynamics of neurons in interesting and sometimes counter-intuitive ways [4], [5], [6]. Thus, a full understanding of these neural firing behaviors requires the analysis of more detailed models.
The effects of dendritic tree architecture on the electrical activity of neurons were first explored by Rall, who demonstrated that each segment of the tree can be effectively modeled as a one-dimensional cable [7], [8], [9]. However, mathematical analysis on the resulting system of coupled one-dimensional partial differential equations quickly becomes intractable once nonlinear ionic currents are included in the dendrites. In this case, various computational studies have shown that the interaction of dendritic topology and the nonlinear ionic currents affect both the firing frequency and the type of firing (regular, bursting) in neurons, e.g., [2], [3], [10]. Alternatively, when the dynamics of each of the dendritic tree segments is linear or quasi-linear, a Green’s function can be derived which captures how the architecture of the tree affects the filtering of a current stimulus input to the dendrite [11], [12], [13]. Although the Green’s function approach allows for a large reduction in computational complexity in exploring how the membrane potential of dendritic trees responds to time-varying inputs, these approaches assume linearity, which prohibits the inclusion of any nonlinear spike-generating mechanisms. As such, Green’s function approaches are limited in their applicability to exploring how dendritic topology affects neuronal firing dynamics.
Recently, Schwemmer and Lewis were able to explore the effects of passive dendritic properties on the dynamics of a leaky-integrate-and-fire (LIF) model neuron [14] that explicitly includes spike effects [6]. Owing to the simplicity of the LIF neuron, they were able to obtain the analytical solution of the system, which they used to show that the inclusion of the dendrite sometimes caused the system to display bistability between periodic oscillations and quiescence, reminiscent of Hodgkins’ type 2 excitability [15]. However, as the dendrite was modeled as a single one-dimensional passive cable, the branching structure of the dendritic tree is ignored. Here, we extend the results of [6] in order to explore how the topology of a passive dendritic tree affects the firing dynamics of the LIF neuron. To accomplish this, we model the dendritic tree as a system of n ordinary differential equations connected by electronic coupling, i.e., each segment of the dendritic tree is now assumed to be a single isopotential compartment [16]. The soma is again modeled using LIF dynamics that explicitly include spike effects. Owing to the simplicity of the system, we are able to obtain the full analytical solution to the -dimensional system, and use it to derive an n-dimensional return map which captures the dynamics of the full system. Using this framework, we seek to understand how dendritic tree properties alter firing dynamics. In particular, we systematically explore how the interaction of dendritic biophysical properties and dendritic tree structure affect the appearance of the bistable behavior that was previously reported in [6]. We find that dendritic topology has a strong quantitative affect on the bifurcation structure of the system, with more complex, branchier dendritic tree topologies tending to promote bistability.
This paper is organized as follows. In Section 2, we describe our multi-compartment LIF model. We then show how one can analytically derive the solution to the system, and use the solution to construct the return map. Using the return map, we systematically explore how biophysical parameters interact with simplified dendritic topologies to affect firing dynamics. We find that dendritic topology has a strong quantitative affect on the bifurcation structure of the system. Lastly, we demonstrate the flexibility of the model by exploring the firing dynamics of a neuron with a more complex dendritic topology.
Section snippets
Multi-compartment leaky-integrate-and-fire model
We model a neuron as an isopotential somatic compartment electrically coupled to a passive dendritic tree. As opposed to other approaches that have modeled the dendritic tree using a series of coupled passive cables (e.g., [8], [13]) or a single equivalent cylinder (e.g., [6]), we choose to model the tree as a series of coupled isopotential compartments, i.e., a multi-compartment model (e.g., [16], [17], [18]). This greatly simplifies the analysis while still allowing us to explore the effects
Results
We now examine the behavior of the multi-compartment LIF neuron by analyzing the map derived in the previous section. First, we describe the functional forms for the spike shape function h(t) that we use in our analysis. We then determine the parameter values at which the time-independent “quiescent” steady-state ceases to exist and the parameter values at which stable periodic oscillations appear. Similar to [6], we find that the system can display bistability between periodic firing and
Discussion
In this work, we have presented a framework to study how complex dendritic tree structures can affect the firing dynamics of an LIF neuron that explicitly includes spike effects. The dendritic tree is modeled as a system of coupled passive compartments, i.e., a multi-compartment model, which allows for flexibility in exploring complex branching topologies. We obtain the analytical solution and use it to derive a lower dimensional return map which completely characterizes the firing dynamics of
Acknowledgments
M.A.S. is supported in part by the Mathematical Biosciences Institute and the National Science Foundation under grant DMS 0931642.
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