Variability of collective dynamics in random tree networks of strongly coupled stochastic excitable elements

We study the collective dynamics of strongly diffusively coupled excitable elements on small random tree networks. Stochastic external inputs are applied to the leaves causing large spiking events. Those events propagate along the tree branches and, eventually, excite the root node. Using Hodgkin-Huxley type nodal elements, such a setup serves as a model of sensory neurons with branched myelinated distal terminals. We focus on the influence of the variability of tree structures on the spike train statistics of the root node. We present a statistical description of random tree networks and show how the structural variability translates into the collective network dynamics. In particular, we show that in the physiologically relevant case of strong coupling the variability of collective response is determined by the joint probability distribution of the total number of leaves and nodes. We further present analytical results for the strong-coupling limit in which the entire tree network can be represented by an effective single element.

Figure 1

Examples of connectivity of nodes of Ranvier in myelinated branching terminals of muscle spindle afferents [(a)–(c)] from [23, 24] and of a touch receptor afferent (d) from [25]. Red semicircles represent heminodes (leaf nodes) which receive external inputs. Internal nodes of Ranvier are marked by blue circles; the green circle marks the root node.

Figure 2

Sufficiently strong input current to leaf nodes fires up the root node. (a) A sample tree with $\mathcal{N}=17$ nodes. The $\mathcal{H}=8$ leaf nodes are marked by red semicircles. The root node is marked by the green circle. Star symbols point at “recording” sites of voltage traces shown in Fig. 3. The shown tree structure reproduces a reconstruction of an experimentally observed muscle spindle afferent neuron presented in Ref. [24]. (b) Threshold current, $J_{\text{th}}$, for the onset of repetitive firing of the root node, as a function of the coupling strength, $\kappa$, for the tree shown in panel (a). The dashed horizontal line marks the theoretical estimate of the threshold current in the strong-coupling limit, $J_{\infty }=(\mathcal{N}/\mathcal{H})J_{\text{AH}}=61.75\mu \mathrm{A}/{\mathrm{cm}}^2$; see in Sec. 3b.

Figure 3

Stochastic dynamics of excitable elements coupled on the tree network shown in Fig. 2. (a) Time traces of the membrane potentials of two leaf nodes and the root marked by stars in Fig. 2, for the indicated coupling strengths. (b), (c) Firing rate, $r$, and coefficient of variation, ${\mathcal{C}}_{\tau }$, for spike trains generated at the root node as a function of coupling strength. Dashed lines in panels (b) and (c) refer to theoretical estimates of the firing rate and CV in the strong-coupling limit. The parameters of input currents to leaves are $J=50\mu \mathrm{A}/{\mathrm{cm}}^2$, $S=500$ ${(\mu \mathrm{A}/{\mathrm{cm}}^2)}^2\text{ms}$.

Figure 4

Tree networks for all 25 possible configurations of 2-tuples of $({\mathcal{D}}_3,{\mathcal{D}}_4)$ resulting for the branching PMF given by Eq. (12) for $G=4$. Leaves are marked red, the root node is marked green, and internal nodes are depicted as blue circles. Trees are arranged in columns with the same total number of leaves and nodes, $(\mathcal{H},\mathcal{N})$, shown on the bottom. The ratio of the total number of nodes to leaves, $\mathcal{N}/\mathcal{H}$, increases from left to right, from 1.75 to 1.9375, respectively.

Figure 5

Node statistics of full binary trees with the branching PMF given by Eq. (12) with $p_0=0.5$ and $G=4$. (a) The joint probability mass function of numbers of nodes in third and fourth generation, $P_2({\mathcal{D}}_3,{\mathcal{D}}_4$). All 25 connectivity states are shown by filled circles. Probabilities for realizing the respective trees are indicated by circle diameter and color. (b) The PMF of respective combinations of total number of leaf nodes and nodes given by Eq. (14).

Figure 6

Realizations and statistics for general binary trees with the branching PMF (15) and maximal allowed number of generation, $G=3$. (a) A sample of 12 different realizations of general binary trees. (b) The PMF of the total number of nodes and leaves, $P_2(\mathcal{H},\mathcal{N})$; 28 possible trees with distinct total numbers of nodes and leaves are shown by filled circles, whose diameter and color represent probability values.

Figure 7

Heat map of the firing rate of a single HH node vs input current and noise intensity, $\tilde{r}(J_{\text{eff}},S_{\text{eff}})$. Symbols mark combinations of $J_{\text{eff},k}$ and $S_{\text{eff},k}$ (17) of realizations of binary trees for the indicated ensembles. The magenta symbols (circles and squares) represent all possible full binary trees with branching PMF (12) and $G=4$. Black star symbols mark all possible general binary trees with branching PMF (15) and $G=3$. Parameters: $S=500$ ${(\mu \mathrm{A}/{\mathrm{cm}}^2)}^2\text{ms}$; $J=38.5\mu \mathrm{A}/{\mathrm{cm}}^2$ (unfilled stars and circles) and $J=70\mu \mathrm{A}/{\mathrm{cm}}^2$ (filled stars and squares).

Figure 8

Normalized threshold current, ${\mathcal{R}}^{-1}(\kappa ,\left\{{\mathcal{D}}_k\right\},\left\{h_k\right\})=J_{\text{th}}(\kappa ,\left\{{\mathcal{D}}_k\right\},\left\{h_k\right\})/J_{\text{AH}}$, as a function of coupling strength $\kappa$ for two tree network ensembles. (a) Curves for all 25 full binary trees of Fig. 4. (b) Same for 50 general binary trees with all possible numbers of nodes and leaf nodes, with maximal number of generations, $G=3$. Curves are color-coded according to the tree's normalized threshold currents in the strong-coupling limit, $R_{\infty ,k}^{-1}={\mathcal{N}}_k/{\mathcal{H}}_k$. Dashed horizontal lines show the value of the applied constant current, $J=38.5\mu \mathrm{A}/{\mathrm{cm}}^2$, used for stochastic simulations in Fig. 9. (c), (d) Normalized threshold current for $\kappa =1000\mathrm{mS}/{\mathrm{cm}}^2$ versus its theoretical strong-coupling limit, $R_{\infty ,k}^{-1}={\mathcal{N}}_k/{\mathcal{H}}_k$ for full (c) and general (d) binary trees.

Figure 9

Spike train statistics of ensembles of excitable trees. The upper panels [(a), (b), (c)] show results for all 25 full binary trees of Fig. 4; the lower panels [(d), (e), (f)] refer to 50 general binary trees with at most $G=3$ generations and with all possible numbers of nodes and leaf nodes. Trees' firing rates [(a), (d)] and CVs [(b), (e)] as functions of coupling strength. Lines are color-coded according to the scaling factor, ${\mathcal{R}}_{\infty ,k}={\mathcal{H}}_k/{\mathcal{N}}_k$, for the strong-coupling limit; dashed horizontal lines show lower and upper limits of corresponding quantities, obtained from the strong-coupling approximation. (c), (f) Firing rate as a function of the CV for $\kappa =1000\mathrm{mS}/{\mathrm{cm}}^2$ for simulated trees (crosses) and for single root nodes (circles) with input current rescaled according to (17). The parameters for numerical simulations are $J=38.5\mu \mathrm{A}/{\mathrm{cm}}^2$, $S=500$ ${(\mu \mathrm{A}/{\mathrm{cm}}^2)}^2\text{ms}$.

Figure 10

Ensemble-averaged ISI statistics as a function of coupling strength for the indicated values of the zero-branching probability. (a), (b) Ensemble-averaged firing rate, $\langle r\rangle$, and CV $\langle {\mathcal{C}}_{\tau }\rangle$ for full binary tress with $G=4$. (c), (d) $\langle r\rangle$ and $\langle {\mathcal{C}}_{\tau }\rangle$ for general binary trees with $G=3$. Dashed lines show predictions of the strong-coupling theory, obtained by ensemble averaging of corresponding values for isolated single nodes with the effective current and noise intensity according to Eq. (17). Parameters: $J=38.5\mu \mathrm{A}/{\mathrm{cm}}^2$, $S=500$ ${(\mu \mathrm{A}/{\mathrm{cm}}^2)}^2\text{ms}$.

Figure 11

Variability of the firing rate due to structural variability. Upper and lower panels show the ensemble-averaged firing rate, $\langle r\rangle$, and its normalized SD, ${\mathcal{C}}_r$, Eqs. (11) and (10), as a function of the probability of zero branching, $p_0$, for the full binary trees and for general binary trees, respectively. Solid lines show results of direct simulations of trees; dashed blue lines show prediction of the strong-coupling theory. Other parameters are the same as in Fig. 10.

Figure 12

Response curves, $r\left(J\right)$, for tree networks with all possible combinations of numbers of leaf nodes and nodes in respective tree ensembles, in the strong-coupling limit. Panel (a) shows results for all possible 25 full binary trees of Fig. 4; panel (b) refers to the 50 general binary trees with at most $G=3$ generations. Lines are color-coded according to the scaling factor, ${\mathcal{R}}_{\infty ,k}={\mathcal{H}}_k/{\mathcal{N}}_k$. Choices of other parameters are the same as in Fig. 10.

Figure 13

Normalized standard deviation of the distribution of root nodes firing rates, ${\mathcal{C}}_r={\mathcal{C}}_r(p_0,J)$, among ensembles of full (a) and general (b) binary tries as a function of the constant input current to leaf nodes and the zero-branching probability for the strong-coupling limit. Other parameters are the same as in Fig. 10.

Figure 14

Firing rate statistics of nonbinary random trees with uniform branching according to the PMF Eq. (20). (a) Three examples of tree networks; pairs of total number of nodes and leaf nodes are (49,70) for the upper panel, (4,10) for the lower left, and (24,40) for the lower right panel. (b) Heat map of the firing rate of a single HH node as a function of input current and noise intensity, $\tilde{r}(J_{\text{eff}},S_{\text{eff}})$. Symbols mark combinations of effective parameters, $J_{\text{eff},k}$ and $S_{\text{eff},k}$ in Eq. (17), for the ensemble of 2000 network realizations. (c) Probability distributions of the firing rate of the root nodes for the indicated values of coupling strengths. For strong coupling, $\kappa =1000\mathrm{mS}/{\mathrm{cm}}^2$, solid and dashed lines compare direct simulations of tree realizations and simulations of effective single nodes, respectively.

Figure 15

Example of a random tree network with $G=4$ (a) and corresponding adjacency matrix (b), constructed following Eqs. (A1) and (A2).

Figure 16

Illustration of notations for the voltage differences, $\mathrm{\Delta }{V}_{j,m_j}$, $\mathrm{\Delta }{V}_{o_j,j}$, and $\mathrm{\Delta }{V}_{o{o}_j,o_j}$, for a tree fragment. Therein $o_j\in (g+1)$ labels the node under consideration; the node with the index $j\in g$ is the parent of $o_j$. The node with the index $o_j\in (g+1)$ is the offspring of $j$ and is the parent of the node $o{o}_j\in (g+2)$. Red semicircles show leaves at which branching is terminated.