Visualizing neuronal network connectivity with connectivity pattern tables

Abstract

Complex ideas are best conveyed through well-designed illustrations. Up to now, computational neuroscientists have mostly relied on box-and-arrow diagrams of even complex neuronal networks, often using ad hoc notations with conflicting use of symbols from paper to paper. This significantly impedes the communication of ideas in neuronal network modeling. We present here Connectivity Pattern Tables (CPTs) as a clutter-free visualization of connectivity in large neuronal networks containing two-dimensional populations of neurons. CPTs can be generated automatically from the same script code used to create the actual network in the NEST simulator. Through aggregation, CPTs can be viewed at different levels, providing either full detail or summary information. We also provide the open source ConnPlotter tool as a means to create connectivity pattern tables.

Keywords: connectivity; neuronal network; population; projection; visualization.

Figure 1

Hierarchy of diagrams of a complex network model (after Hill and Tononi, 2005). (A) Overview diagram showing only the main parts of the network, i.e., retina (Ret.), thalamus (Tp), thalamic reticular nucleus (Rp), and cortical populations Vp(v) and Vp(h), tuned to vertically and horizontally oriented stimuli, respectively. Arrows mark excitatory, circles inhibitory connections. (B) Detailed diagram of connectivity within cortical population Vp(v). Vp(v) is composed of three cortical layers, each with an excitatory (left) and inhibitory (right) subpopulation. Solid lines represent excitatory, dashed lines inhibitory connections. (C) Detailed rendition of connection masks and kernels projecting onto one cortical subpopulation Vp(v)L56(e) from panel (B), i.e., the excitatory subpopulation of layer 5/6 of Vp(v). Modified from Nordlie et al. (, Figure 6).

Figure 2

Example network and connectivity matrices. (A) A simple network consisting of four parts a, b, c and d. The intensity I of each connection is marked. Connectivity matrices, based on (B) CoCoMac (http://cocomac.org), indicating the existence of connections by 0 and 1, (C) Binzegger et al. (2004), indicating connection strength by color code ranging from dark red to white, and (D) Hinton-diagrams (Hinton et al., 1986), indicating connection type and strength by color and area, respectively. In all matrices, sources are represented by rows, targets by columns.

Figure 3

CPTs for the Simple network connectivity defined in Table 1. Rows represent sources, columns targets. Excitatory connections are shown in red, inhibitory ones in blue. Saturation reflects the connection intensity defined as product of connection probability and synapse weight in arbitrary units. (A) Local color scale for each patch; patches with only a single value show fully saturated color. (B) Global color scale; only the maximum of the Gaussian RG/E→RG/I connection reaches full saturation. This CPT is discussed in detail in the section “Simple network CPT”.

Figure 4

CPT for the Complex network connectivity defined in Table 2. Full table showing the four different synapse types (AMPA: red, NMDA: orange, GABA_A: blue, GABA_B: purple) using (A) local and (B) a global color scale. This CPT is discussed in detail in the section “Complex network CPT”.

Figure 5

Aggregated CPTs for the Simple network. (A) Connections for each source/target pair of population groups are aggregated, but synapse types kept separate. (B) Connections aggregated across layers and synapse types. As for all CPTs aggregating across synapse types, the color scale ranges from blue (most inhibitory) via white (neutral) to most excitatory (red). (C) As in panel (B), but now using a global scale for colors in all patches. The arrow at the right end of the color bar indicates that the CPT contains values >5.

Figure 6

Aggregated CPTs for the Complex network show in Figure 4. (A) Connectivity aggregated across groups, but not synapse types. (B) Connectivity aggregated across synapse types for each source/target pair of populations, i.e., combining either AMPA and NMDA or GABA_A and GABA_B. (C) Connectivity aggregated across groups and synapse types.

Figure 7

Full CPT for the Hill-Tononi model reduced to the primary pathway (Hill and Tononi, 2005). Layers are retina (Ret), thalamus (Tp; two populations: relay cells and interneurons), reticular nucleus (Rp), and horizontal and vertically tuned primary visual cortex (Vp_h, Vp_v); the latter two layers have six populations each, representing pyramidal cells and interneurons in layers 2/3, 4, and 5/6. Synapse types are AMPA, NMDA, GABA_A, and GABA_B, with the same color code as in Figure 4. Note the rectangular projections from population Tp/Relay to Vp_h and Vp_v.

Figure 8

Aggregated CPTs for the Hill-Tononi model from Figure 7. (A) Connectivity aggregated across synapse types for each population pair. (B) Connectivity combined across populations in each group, but not across synapse types. (C) As in panel (B), but also aggregated across synapse types. (D) As in panel (C), but with a global color scale limited to [−2,2].

Figure 9

Aggregated CPTs based on total charge deposited. The CPTs shown here are equivalent to Figure 8D, but intensity is now I_tcd defined in terms of total charge deposited in femtocoulomb, cf. Eq. 4 for membrane potential (A) V_m = −70 mV and (B) V_m = −50 mV, respectively. The color scale is the same for both figures. Note the difference in mid-range inhibition in intracortical connections.