Symmetry group factorization reveals the structure-function relation in the neural connectome of Caenorhabditis elegans

Abstract

The neural connectome of the nematode Caenorhabditis elegans has been completely mapped, yet in spite of being one of the smallest connectomes (302 neurons), the design principles that explain how the connectome structure determines its function remain unknown. Here, we find symmetries in the locomotion neural circuit of C. elegans, each characterized by its own symmetry group which can be factorized into the direct product of normal subgroups. The action of these normal subgroups partitions the connectome into sectors of neurons that match broad functional categories. Furthermore, symmetry principles predict the existence of novel finer structures inside these normal subgroups forming feedforward and recurrent networks made of blocks of imprimitivity. These blocks constitute structures made of circulant matrices nested in a hierarchy of block-circulant matrices, whose functionality is understood in terms of neural processing filters responsible for fast processing of information.

Fig. 1

Group theoretical definitions: automorphism, symmetry groups, pseudosymmetries, normal subgroups, and blocks of imprimitivity. a Circuit made of gap-junction and only interneurons in the forward locomotion used to define an automorphism. These are permutation symmetries that leave the adjacency structure invariant. These symmetries then convert to a system of imprimitivity when we integrate the circuit into the full locomotion connectome. Nodes represent neurons and weighted links represent the number of gap-junctions connections between neurons from ref. . b Adjacency matrix of the circuit in a. This matrix is composed of circulant matrices: a high-pass filter $\mathcal{H}=\mathrm{circ}\left(0,1\right)$ in the diagonal and an off-diagonal low-pass filter $\mathcal{L}=\mathrm{circ}\left(1,1\right)$. The full $4\times 4$ matrix forms a block-circulant matrix $\mathcal{B}C=\mathrm{bcirc}\left(\mathcal{H},\mathcal{L}\right)$ (see Methods Section for definitions). c Symmetry group of the circuit shown in a, called dihedral group ${\mathbf{D}}_{\mathbf{8}}$, comprises 8 automorphisms out of the 4! = 24 possible permutations of neurons. We show each permutation matrix $P$ of each automorphism. d Pseudosymmetries capture inherent variabilities in the connectome from animal to animal. An example pseudosymmetry is shown $P_{\epsilon }=$ DB5 $\leftrightarrow$ DB6 that breaks one link to AVBR over 18 total weighted links, giving $\epsilon =1∕18=5.5\%$. e Definition of normal subgroup. A subgroup $\mathbf{H}$ is said to be normal in a group $\mathbf{G}$ if and only if $\mathbf{H}$ commutes with every element $g\in \mathbf{G}$, that is: $\left[g,\mathbf{H}\right]=g\mathbf{H}-\mathbf{H}g=0$ (see Supplementary Note 4 for a detailed explanation). f Definition of blocks of imprimitivity and system of imprimitivity. Simply put, a set of nodes is called a block (of imprimitivity) if all nodes in this set always ‘move together’ under any automorphism of the symmetry group. A set of blocks with such a property is thus called a system of imprimitivity (see Supplementary Note 7 for a formal definition). g Definition of circulant matrix and circular convolution. Matrix $\mathcal{F}$ appears in the forward gap-junction locomotion circuit and is called a circulant matrix. This matrix has a peculiar pattern where each row is a shift to the right by one entry of the previous row. Multiplication of $\mathcal{F}$ by a vector $\mathbf{x}$ gives rise to a famous operation called a circular convolution, which is used in many applications, ranging from digital signal processing, image compression, and cryptography to number theory, theoretical physics and engineering, often in connection with discrete and fast Fourier transforms, as explained in Supplementary Note 8

Fig. 2

Symmetry group ${\mathbf{F}}_{\mathbf{gap}}$ of the forward gap-junction circuit. a Circuit from ref. . Pseudosymmetries $P_{\epsilon }$ act on distinct sectors of neurons indicated by different colors that lead to direct product factorization of the symmetry group into normal subgroups. The normal subgroups sectors of neurons match the broad classification of command interneurons and motor neurons from the Wormatlas. b Adjacency matrix of (a) showing the normal subgroup structure and its matching with broad neuronal classes. c Idealization of the circuit obtained from a by $\epsilon \to 0$ leading to perfect symmetries (see Supplementary Note 3). We highlight the two 4-cycles across ${\mathcal{B}}_1$: VB2 $\to$ DB3 $\to$ DB2 $\to$ VB1 $\to$ VB2 and its conjugate ${\mathcal{B}}_2$: DB1 $\to$ VB4 $\to$ VB5 $\to$ VB6 $\to$ VB4 that give rise to the circulant matrix structure highlighted in the checker-board pattern in d of both imprimitive blocks. d Adjacency matrix of the ideal circuit in c. We highlight the two imprimitive blocks ${\mathcal{B}}_1=$ (VB2, DB3, DB2, VB1) and ${\mathcal{B}}_2=$ (DB1, VB4, VB5, VB6) mentioned in the text and its circulant structure in the normal subgroup ${\mathbf{D}}_{\mathbf{1}}$. The other normal subgroups are also described by circulant blocks and correspond to imprimitive blocks: ${\mathcal{B}}_3,$${\mathcal{B}}_4,$${\mathcal{B}}_5,$${\mathcal{B}}_6,$${\mathcal{B}}_7$, as indicated. Some of these structures also form block-circulant matrices. Each block of the adjacency matrix $A$ performs a fundamental signal processing task

Fig. 3

Symmetry group ${\mathbf{B}}_{\mathbf{gap}}$ of the backward gap-junction circuit. a The real circuits and (b) its adjacency matrix. The symmetry group is factorized as a direct product of normal subgroups: ${\mathbf{B}}_{\mathbf{gap}}=\left[{\mathbf{C}}_{\mathbf{2}}\times {\mathbf{C}}_{\mathbf{2}}\times {\mathbf{D}}_{\mathbf{1}}\right]\times \left[{\mathbf{S}}_{\mathbf{12}}\times {\mathbf{D}}_{\mathbf{6}}\times {\mathbf{C}}_{\mathbf{2}}\right]$, which leads to a partition of neurons in two sectors that match the command and motor sectors known experimentally, as indicated. c Ideal circuit and (d) adjacency matrix highlighting the primitive and imprimitive blocks and their circulant structures from ${\mathcal{B}}_1$ to ${\mathcal{B}}_9$

Fig. 4

Symmetry groups ${\mathbf{F}}_{\mathbf{ch}}$ and ${\mathbf{B}}_{\mathbf{ch}}$ of the chemical synapse forward and backward circuits. a Forward locomotor chemical synapse circuit and (b) its adjacency matrix (ideal circuits, real circuits in Supplementary Figs. 5 and 6). The symmetry group ${\mathbf{F}}_{\mathbf{ch}}$ is factorized into the direct product of command, motor, and touch subgroups as ${\mathbf{F}}_{\mathbf{ch}}={\mathbf{C}}_{\mathbf{2}}\times \left[{\mathbf{D}}_{\mathbf{1}}\right]\times \left[{\mathbf{S}}_{\mathbf{10}}\times {\mathbf{D}}_{\mathbf{1}}\right]$, which, in turn, split up the circuits into independent sectors of neurons matching different functions and include also the neuron touch class PVC (forward) and AVD (backward). c The backward circuit factorizes as ${\mathbf{B}}_{\mathbf{ch}}={\mathbf{C}}_{\mathbf{2}}\times \left[{\mathbf{C}}_{\mathbf{2}}\times {\mathbf{C}}_{\mathbf{2}}\right]\times \left[{\mathbf{S}}_{\mathbf{5}}\times {\mathbf{S}}_{\mathbf{4}}\times {\mathbf{S}}_{\mathbf{3}}\times {\mathbf{D}}_{\mathbf{1}}\times {\mathbf{C}}_{\mathbf{2}}\times {\mathbf{C}}_{\mathbf{2}}\right].$ We show the ideal circuit and (d) its adjacency matrix. For simplicity we plot only the interneurons that connect to the motor neurons. Full circuit in SM Fig. 6. All neurotransmitters are cholinergic and excitatory (ACh) except for RIM which uses neurotransmitter Glutamate and Tyramine and AIB which is glutamatergic (see Supplementary Note 6). These different types of synaptic interactions respect the symmetries of the circuits, see Supplementary Note 5

Fig. 5

Symmetry vs. other methods. We compare the functional classes obtained from symmetries with modularity detection algorithms^, and a typical eigenvector centrality measure. a Forward gap-junction circuit classes obtained using modularity or community detection detection algorithm from ref. and (b) using eigenvector centrality. c Backward gap-junction circuit modularity and (d) eigenvector centrality. Both measures, modular detection and centrality, do not capture the symmetries and functional classification of this connectome