Emergent Network Topology within the Respiratory Rhythm-Generating Kernel Evolved <i>In Silico</i>

<div><p>We hypothesize that the network topology within the pre-Bötzinger Complex (preBötC), the mammalian respiratory rhythm generating kernel, is not random, but is optimized in the course of ontogeny/phylogeny so that the network produces respiratory rhythm efficiently and robustly. In the present study, we attempted to identify topology of synaptic connections among constituent neurons of the preBötC based on this hypothesis. To do this, we first developed an effective evolutionary algorithm for optimizing network topology of a neuronal network to exhibit a ‘desired characteristic’. Using this evolutionary algorithm, we iteratively evolved an <i>in silico</i> preBötC ‘model’ network with initial random connectivity to a network exhibiting optimized synchronous population bursts. The evolved ‘idealized’ network was then analyzed to gain insight into: (1) optimal network connectivity among different kinds of neurons—excitatory as well as inhibitory pacemakers, non-pacemakers and tonic neurons—within the preBötC, and (2) possible functional roles of inhibitory neurons within the preBötC in rhythm generation. Obtained results indicate that (1) synaptic distribution within excitatory subnetwork of the evolved model network illustrates skewed/heavy-tailed degree distribution, and (2) inhibitory subnetwork influences excitatory subnetwork primarily through non-tonic pacemaker inhibitory neurons. Further, since small-world (SW) network is generally associated with network synchronization phenomena and is suggested as a possible network structure within the preBötC, we compared the performance of SW network with that of the evolved model network. Results show that evolved network is better than SW network at exhibiting synchronous bursts.</p></div

Network activity before and after evolution using MM1 and, subsequently using MM3.

<p>Network is evolved to minimize (Cost1+Cost2) as defined using Eqs (<a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0154049#pone.0154049.e001" target="_blank">1</a>) and (<a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0154049#pone.0154049.e002" target="_blank">2</a>). (A)‘Initial’: Raster plot depicting activities of 80 neurons depicted in <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0154049#pone.0154049.g005" target="_blank">Fig 5A</a> when they are randomly synaptically connected with <i>SynFrac</i> = 0.10. (A)‘Evolved using MM1’: Raster plot depicting activities of same 80 neurons when the network connectivity is evolved to optimum using MM1; <i>SynFrac</i> is maintained fixed at 0.10. This evolved network is then further evolved using MM3 and its performance is shown in (A)‘Further evolved using MM3’. (B) Solid curve: Time-history of total number of bursting neurons at a given instant. Dashed horizontal line: Threshold level, set at 30% of 80 neurons (our adopted convention), which when exceeded by ‘solid curve’, the network is considered to be exhibiting a population burst. (C) Population activity (averaged membrane potential) of all 80 neurons.</p

Classification of <i>g</i>_<i>Leak</i>/<i>g</i>_<i>NaP</i>-parametric space into three regions—Tonic, Core and Activated.

<p>Distribution of 80 neurons constituting the neuronal network over <i>g</i>_<i>Leak</i>/<i>g</i>_<i>NaP</i>-parametric space is also indicated and is identical with that depicted in <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0154049#pone.0154049.g005" target="_blank">Fig 5C</a>. See section 3.2.1 for further explanation.</p

Composite network activity before and after evolution using MM3.

<p>Network is evolved to minimize the cost (Cost1+Cost2+Cost3), see <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0154049#pone.0154049.g011" target="_blank">Fig 11</a>. (A)‘Initial’: Raster plot depicting activities of 160 neurons—first 80 are excitatory neurons while the rest are inhibitory neurons—within the neuronal network when they are randomly synaptically connected with <i>SynFrac</i> = 0.06; (A)‘Evolved using Simulated annealing’: Raster plot depicting activities of same 160 neurons when the network connectivity is evolved using simulated annealing. <i>SynFrac</i> value varies during evolution process (see <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0154049#pone.0154049.g011" target="_blank">Fig 11</a>); final <i>SynFrac</i> ≈ 0.12. (A)‘Random Network’: Raster plot depicting activities of 160 neurons in a random network with <i>SynFrac</i> ≈ 0.12. (B) Solid curve: Time-history of total number of bursting neurons at a given instant. Dashed horizontal line: Threshold level, 30% of 80 excitatory neurons (our adopted convention), which when exceeded by ‘solid curve’, the network is considered to be exhibiting a population burst. (C) Population activity (averaged membrane potential) of all 80 excitatory neurons.</p

Evolution of small-world (SW) network.

<p>(A,B & C) Typical neuronal activity of a SW network, initially and after evolving it using MM4. Objective of evolution is to minimize (Cost1+Cost2) as defined by Eqs (<a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0154049#pone.0154049.e001" target="_blank">1</a>) and (<a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0154049#pone.0154049.e002" target="_blank">2</a>), respectively. (D) Evolution of cumulative curves (CC) depicting participation level of individual neurons to each population burst. CC_objective: CC corresponding to the desired case when every neuron in the network bursts with every population burst. CC_MM4 is CC corresponding to network activity depicted in (A)‘Evolved SW using MM4’. CC₀, CC₁, CC₂ and CC₃ are representative intermediate CC obtained during the evolution process. CC_MM3 is CC corresponding to evolved excitatory network by method described in section 2.5 (it is the same CC_MM3 curve as is depicted in <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0154049#pone.0154049.s008" target="_blank">S8 Fig</a>).</p

Synaptic distribution among network neurons before and after evolution (using MM3) corresponding to the case presented in Fig 6.

<p>(A) Initial incoming synaptic distribution, (B) Initial outgoing synaptic distribution, (C) Outgoing synaptic distribution after evolution, and (D) Incoming synaptic distribution after evolution. Neurons indicated in square shape with thick white borders (consequently, appearing smaller in size) towards the bottom-right region in each panel are those which remain silent in the ‘evolved’ network activity, <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0154049#pone.0154049.g006" target="_blank">Fig 6</a>. (E)&(F) Connectivity of top three richest neurons in (C)&(D), respectively, with respect to rest of the neurons in the network. The top three richest neurons in (C)&(D) are indicated by magenta colored arrows for convenience.</p

Method concept.

<p>For a mathematical model neuronal network with fixed intrinsic parameters of constituting neurons, its group behavior is a function of network topology alone. Different connection topology of synapses among neurons of network produce different group behavior. Group behavior may be visualized in terms of a raster plot from which essential feature may be subsequently extracted in the form of a ‘characteristic curve’. By employing an evolutionary algorithm, network topology is iteratively evolved so that its characteristic curve matches a ‘desired characteristic’.</p

Figure illustrating usage of adjacency matrix for handling network connectivity.

<p><b>(A)</b> Neuronal network consisting of 5 excitatory and inhibitory neurons, represented by tan and blue colors, respectively. <b>(B)</b> Adjacency matrix depicting the connectivity among neurons in network shown in (A). For example, red synapse connecting neuron 1 to neuron 3 in (A) is identified by presence of 1 at row 1 and column 3 in adjacency matrix. <b>(C)</b> Four partitions of adjacency matrix based on the nature of neuron, excitatory (represented by E) or inhibitory (represented by I) at its pre- and post-synaptic terminals. For example, partition E-I of adjacency matrix contains information of synapses emerging from excitatory neurons and terminating on inhibitory neurons.</p

Histograms indicating distribution of neurons with various degrees of out-going and incoming synapses in the evolved network depicted in Fig 7.

<p>Histograms indicating distribution of neurons with various degrees of out-going and incoming synapses in the evolved network depicted in <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0154049#pone.0154049.g007" target="_blank">Fig 7</a>.</p

Synaptic distribution among neurons of the evolved network corresponding to the case presented in Fig 12.

<p>Neurons indicated in square shape with thick white borders (and consequently, appearing smaller and fainter) in each panel are those which remain silent in the ‘evolved’ network activity, <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0154049#pone.0154049.g012" target="_blank">Fig 12</a>. Initial synaptic distribution among neurons is randomly distributed with overall <i>SynFrac</i> = 0.06, which is lower than that of evolved network (<i>SynFrac</i> ≈ 0.12) and hence is not shown.</p