Selective inhibition as a subspace rotation.

(a) Schematic of a recurrent network of excitatory (red) and inhibitory neurons (green) for sequence generation. A second population of inhibitory neurons (blue) is responsible for sequence selection. Black arrows denote excitatory input. (b) Simplified circuit diagram of three excitatory neurons, with neuron 2 receiving selective inhibition. For this example, recurrent interactions are neglected. (c) Schematics of firing rates of the selective inhibitory input affecting the excitatory neuron 2, top, and the response of the three excitatory neurons below, as shown in b. Three cases for the inhibitory input onto neuron 2 are shown. A baseline condition as well as strong and weak inhibition. Strong inhibitory input silences neuron 2 and weak input lowers the firing rate. (d,e) Joint activity of the neurons resides on a two dimensional subspace. Colors and numbering correspond to cases in c. (d) Inhibition rotates the subspace. (e) Silencing neuron 2 rotates the circuit subspace onto the plane spanned by neurons 1 and 2.

Selective inhibition preserves sequence generation.

(a) Schematic of network connectivity. Excitatory connections (red) on the ring are local and asymmetric while inhibition (green) is global, leading to sequence propagation. Selective inhibition (blue) targets neurons randomly along the ring. (b) Connectivity matrix W, left, and an example projection matrix P, right. Only 50x50 neurons are shown for visualization. (c) Activity bump moving on ring for no projection (W), left, and a subspace projection (PW), right. Example shows fraction silenced pinh = 0.6. (d) Example eigenvalue spectrum for W, red, and PW, blue, with fraction silenced pinh = 0.6. (e) Maximum eigenvalue as a function of fraction silenced for W and PW. (f) Principal eigenvector for W and PW for same example, pinh = 0.6. (g) Radius of real part and imaginary part of normalized principal eigenvector as a function of fraction silenced for PW, blue lines. Normalized principal eigenvector for W lies on the unit circle, shown for reference in red. (h) Percent change in instantaneous speed, width, and amplitude of the activity bump as a function of fraction silenced. In e,g,h line shows mean and shaded region shows standard deviation over 10 different projection matrices.

Selective inhibition projects activity onto unique neural subspaces.

(a) Schematic of ring network for sequence generation and two inhibitory ensembles for sequence selection via projections onto specific subspaces. (b) Projections onto PC space for four examples of pinh. Dark grey line shows in-subspace and colored lines show out-of-subspace projections. (c) Projection magnitude of out-of-subspace projections and principal angle between subspaces as a function of fraction silenced. Pairwise comparisons for 10 different inhibitory ensembles, i.e. projection matrices P. Lines show mean and shaded regions standard deviation over the 45 possible pairs for each fraction silenced. (d) Proportion overlap between active neurons in pairs of subspaces as a function of fraction silenced, when n = 10 to n = 1000 subspaces were stored. Black line shows mean for n = 1000 and shaded region standard deviation. Colored lines show maximum overlap over all pairs of subspaces for each number of stored subspaces. (e) Probability distribution 𝕡 of two subspaces overlapping by ρ for different fractions of silenced neurons.

A neural circuit for dynamic subspace selection.

(a) Schematic of network structure for sequence selection via dynamic rotation of subspaces with selective inhibition. Colors same as previous figures, top down signal, black arrows. Two inhibitory ensembles compete via winner-take-all dynamics to project activity onto their selected subspace. (b) Circuit motif underlying connectivity shown in (a). (c) Projections of activity from (f) onto neural subspaces. Color code distinguishes periods of time when inhibitory ensemble #1 was active, blue, vs. inhibitory ensemble #2, red. (d) Top down signal to selective inhibitory assemblies. (e) Activity of selective inhibitory assembles. (f) Activity of neurons in the ring network. (g) Projection magnitude in subspace #1 and #2, in blue and red, respectively.

Projection onto neural subspaces in a locally connected spiking neural network for sequence generation.

(a) Each neuron (left, black point) forms local connections based on a Gaussian kernel (schematic: blue circle, excitatory kernel; red circle, inhibitory kernel). Each excitatory neuron projects in a preferred direction (black arrow), defined by an angle θ, implemented as a shift in the center (grey point) of the Gaussian kernel (blue circle). Inhibitory neurons make symmetric projections with a wider Gaussian kernel (red circle). Preferred directions are correlated for nearby neurons (right side, colors denote preferred projection angle θ for excitatory neurons on a 2D grid). (b) Network structure with selective inhibitory ensembles providing spatially heterogeneous, clustered inhibition silencing a fraction pinh of the neurons uniformly distributed in the 2D spiking network. Input region (yellow square) receives excitatory input to evoke sequence. Grid depicts the NE = 14400 excitatory neurons on a 120x120 grid, with landscape of projection asymmetries from (a). (c) Example evoked spatiotemporal activity sequence for fraction silenced pinh = 0.7. Yellow at t = 0 shows input region. Each point represents a spike, every 25th spike is shown. (d) Examples of heterogeneous neuron responses in each of the five different contexts. (e) Variance explained by the first k components for each fraction silenced pinh ∈ [0, 0.9]. (f) Principal angles for the leading 12 principal components (neural modes). Colors denote fraction silenced and each line represents one trial (n = 5 trials). (g) First principal angle as a function of fraction silenced. (h) Histograms demonstrating mixed selectivity for different fraction silenced. Each histogram shows the number of neurons that participated in 0,1,2,3,4 or 5 of the context-dependent spatiotemporal sequences.