Inference of topology and the nature of synapses, and the flow of information in neuronal networks

The characterization of neuronal connectivity is one of the most important matters in neuroscience. In this work, we show that a recently proposed informational quantity, the causal mutual information, employed with an appropriate methodology, can be used not only to correctly infer the direction of the underlying physical synapses, but also to identify their excitatory or inhibitory nature, considering easy to handle and measure bivariate time series. The success of our approach relies on a surprising property found in neuronal networks by which nonadjacent neurons do “understand” each other (positive mutual information), however, this exchange of information is not capable of causing effect (zero transfer entropy). Remarkably, inhibitory connections, responsible for enhancing synchronization, transfer more information than excitatory connections, known to enhance entropy in the network. We also demonstrate that our methodology can be used to correctly infer directionality of synapses even in the presence of dynamic and observational Gaussian noise, and is also successful in providing the effective directionality of intermodular connectivity, when only mean fields can be measured.

Figure 1

Normalized membrane potential of two chemical coupled HR neurons with connection from $x_2$ (red line) to $x_1$ (black line). We consider the coupling strength $g_c=1$ and the mapping time step $\mathrm{\Delta }t=1\mathrm{ms}$. The black circles and red squares correspond to $X_1$ and $X_2$, respectively.

Figure 2

Directional index (DI), causal mutual information (CaMI), and mutual information (MI), as a function of the coupling strength ($g_c$) for the mapping step time $\mathrm{\Delta }t=0.5\mathrm{ms}$ and $L=8$. The red squares, green diamonds, blue triangles, and black circles correspond to $D_{ij}$, $C_{ij}$, $C_{ji}$, and $I_{ij}$, respectively.

Figure 3

(a) Normalized directional index, ranked from larger to smaller values, for a random neuronal network with $N=64$ HR neurons, $K=4$, $\mathrm{\Delta }t=0.25$, $L=8$, $T=4{10}^6$, and $g_c=0.1$. We consider three cases for the connectivity: 256 excitatory synapses (black line), 256 inhibitory synapses (red line), and 128 excitatory and 128 inhibitory synapses (green line). (b) Magnification of (a). (c) Matrix of the normalized directional index ($D_{ij}$) of the latter case.

Figure 4

Normalized directional index for ${\sigma }_d=3$ (black line) and ${\sigma }_d=4$ (green line). We consider $\mathrm{\Delta }t=0.25$, $L=8$, and $g_c=0.15$.

Figure 5

Membrane potential for the neuron $i=32$ for additive noise standard deviation (a) ${\sigma }_a=0.1$ and (b) ${\sigma }_a=0.35$. (c) Normalized directionality index for ${\sigma }_a=0.1$ (black line) and ${\sigma }_a=0.35$ (green line). We consider $\mathrm{\Delta }t=0.5$, $L=4$, and $g_c=0.1$.

Figure 6

Normalized directional index, ordered from larger to smaller values for $N=277$ HR neurons. We consider ${\sigma }_d=1$, $g_c=0.035$, $\mathrm{\Delta }t=0.25$, and $L=8$.

Figure 7

(a) Directionality index (DI) as a function of coupling strength ($g_c$). For $\mathrm{\Delta }t=0.25$ ms, where $L=1$ (black circles), $L=2$ (red squares), $L=4$ (green diamonds), and $L=8$ (blue triangles). Time evolution of the normalized mean field (NMF) for network 1 (black line) and network 2 (red line), both with (b) $g_c=0.025$, (c) $g_c=0.175$, and (d) $g_c=0.275$.