Statistical physics approach to dendritic computation: The excitable-wave mean-field approximation

We analytically study the input-output properties of a neuron whose active dendritic tree, modeled as a Cayley tree of excitable elements, is subjected to Poisson stimulus. Both single-site and two-site mean-field approximations incorrectly predict a nonequilibrium phase transition which is not allowed in the model. We propose an excitable-wave mean-field approximation which shows good agreement with previously published simulation results [Gollo et al., PLoS Comput. Biol. 5, e1000402 (2009)] and accounts for finite-size effects. We also discuss the relevance of our results to experiments in neuroscience, emphasizing the role of active dendrites in the enhancement of dynamic range and in gain control modulation.

Figure 1

Model of an active dendritic tree. (a) A famous drawing by Ramon y Cajal of a human Purkinje cell. (b) Excitable elements (circles) connected (bars) in a Cayley tree topology with $G=2$ layers and coordination number $z=3$ (one mother and $k=2$ daughter branches). Dendritic branchlets are driven by independent Poisson stimuli (small arrows). (c) Each dendritic branchlet can be in one of three states: quiescent (0), active (1), or refractory (2).

Figure 2

Response curve $F\left(h\right)$ for simulations (symbols) and mean-field approximations (lines; 1S, 2S, and black for EW) for $p_{\lambda }=0.7$ and $G=10$. Horizontal and vertical arrows show the relevant parameters for calculating the dynamic range $\Delta$ [see Eq. (1)].

Figure 3

Mean-field approximations. (a) Firing rate of the 1S, 2S, and EW (black) approximations in the absence of stimulus ($h=0$). (b) Family of response functions for $\beta =1$ and $p_{\lambda }=0,0.2,0.4,...,1$. Symbols represent simulations (as in Ref. [20]) and curves are the 1S mean-field approximation. Open (closed) symbols correspond to probabilistic (deterministic, $p_{\lambda }=1$) neighbor coupling, and dotted (continuous) curves correspond to $G=10$ ($G\to \infty$). (c) Phase diagram under the 1S approximation for different system sizes. (d) Same as (b) but for 2S approximation with $G\to \infty$.

Figure 4

Two-site mean-field approximation schematic representation of a general pair ($x$ and $y$) in a binary tree ($k=2$). To describe the dynamics of $x$ and $y$, each of their neighbors must be taken into account. According to Eq. (15), the joint probability of the labeled sites is rewritten in terms of two-site probabilities.

Figure 5

Schematic representation of the excitable-wave mean-field approximation. (a) Dynamics of each layer ($g>0$) in the excitable-wave (EW) mean-field approximation (see text for details). There are three different active states: 1A represents activity coming from an external input; 1B is reached due to forward activity from the next layer, whereas 1C is excited by backpropagating activity from the previous layer. (b) Schematic EW mean-field approximation dynamics in a tree with $G$ layers. Note that there are no loops in the activity flux.

Figure 6

Response functions: Simulations compared to the EW approximation (both with $\beta =1$) and experimental data. (a) Family of response functions for $G=10$ and $p_{\lambda }=0,0.2,0.4,...,1$. Symbols are the simulations (as in Ref. [20]) and solid curves are the EW mean-field approximation. (b) Family of response functions for $p_{\lambda }=0.7$ and different tree sizes: $G=5,10,15,20$. (c) Simulations and EW response function for external stimuli spatially distributed as $h\left(g\right)=h_0{e}^{ag}$: from right to left, $a=0,0.1,0.3,...,0.9$ ($p_{\lambda }=0.8$ and $G=10$). Horizontal lines are plotted for the estimation of the dynamic range. (d) Experimental result from mouse retinal ganglion cells shows double sigmoid response curves (closed symbols) as a function of stimulus $I$ (measured in rhodopsins/isomerizations/rod/second [58]). They can be reasonably well fit by both simulations (open symbols, $p_{\lambda }=0.58$ and $h=0.37$ $I$) and the EW response function ($p_{\lambda }=0.59$ and $h=0.4$ $I$).

Figure 7

Dynamic range as a function of the coupling parameter for 1S ($G=10$), 2S (infinite tree), EW (black lines) mean-field approximations compared to simulations (symbols, as in Ref. [20]) for different trees sizes $G$. The inset shows the dynamic range of dendritic trees subjected to an asymmetrical activity propagation probability controlled by $\beta$ (symbols stand for the simulations and the curves for the EW approximation for $G=10$).

Figure 8

Effects of the priority order of the excitable components on the response functions: simulations compared to the EW approximation and experimental data. Family of response functions for $G=10$ and $p_{\lambda }=0,0.2,0.4,...,1$. Top panels display combinations of 1B prior to 1C, and bottom panels the converse. Symbols are the simulations and solid curves are the EW mean-field approximation.