Recall we are replacing the empirical with a mathematical construct and hope that it is helpful. you see the word dimension tossed around all too freely when the concept of orthogonal is hard to imagine in terms of biology.
Having said that what do we have here. It is the application of the ability of the mathematics to keep independent, but finite inputs separate. All good but also arbitrary.
I do find that the conceptualization is helpful in understanding how our brain works and the use of virtual voids seriously addresses the possibility of accessing past experience however fuzzy.
This is exciting and important because it provides a theoretical framework to successfully hang our empirical understanding upon..
The human brain sees the world as an 11-dimensional multiverse
By Michael Blaustein
June 13, 2017 | 11:28am | Updated
New research suggests that the human brain is almost beyond comprehension because it doesn’t process the world in two dimensions or even three. No, the human brain understands the visual world in up to 11 different dimensions.
The astonishing discovery helps explain why even cutting-edge technologies like functional MRIs have such a hard time explaining what is going on inside our noggins. In a functional MRI, brain activity is monitored and represented as a three-dimensional image that changes over time. However, if the brain is actually working in 11 dimensions, looking at a 3D functional MRI and saying that it explains brain activity would be like looking at the shadow of a head of a pin and saying that it explains the entire universe, plus a multitude of other dimensions.
[ Oh good. i came to understand a long time ago that the whole problem of memory goes away if we simply use wormholes or some such thing to merely go back in time to access the information. The idea of a greater dimensionality allows us to do this by producing a larger theoretical framework, but i suspect it is unnecessary in terms of my cloud cosmology. - arclein ]
The team of scientists led by a group from Scientists at the École Polytechnique Fédérale de Lausanne in Switzerland detected the previously unknown complexities of the brain while working on the Blue Brain Project. The project’s goal is to create a biologically accurate recreation of the human brain.
During their research, the scientists created simulations of the brain and applied an advanced form of mathematics, called algebraic topology, to their computer-generated models.
“Algebraic topology is like a telescope and microscope at the same time. It can zoom into networks to find hidden structures — the trees in the forest — and see the empty spaces — the clearings — all at the same time,” said study author Kathryn Hess.
What Hess and her colleagues found was that the brain processes visual information by creating multi-dimensional neurological structures, called cliques, which disintegrate the instant they are understood, according to Newsweek who first reported on the research that was published in the journal Frontiers in Computational Neuroscience.
The cliques have up to 11 different dimensions and form in holes of space, called cavities. Once the brain understands the visual information, both the clique and cavity disappear.
[ i assume that understand means label - arclein ]
“The appearance of high-dimensional cavities when the brain is processing information means that the neurons in the network react to stimuli in an extremely organized manner,” said researcher Ran Levi.
[ we assume the concept of dimensionality is used here to indicate completely independent inputs which is a good plan. very good - arclein ]
“It is as if the brain reacts to a stimulus by building then razing a tower of multi-dimensional blocks, starting with rods (1D), then planks (2D), then cubes (3D), and then more complex geometries with 4D, 5D, etc. The progression of activity through the brain resembles a multi-dimensional sandcastle that materializes out of the sand and then disintegrates,” he said.
Henry Markram, director of Blue Brain Project, explained just how momentous a discovery the multi-dimensional structures could be.
“The mathematics usually applied to study networks cannot detect the high-dimensional structures and spaces that we now see clearly,” he said.
“We found a world that we had never imagined. There are tens of millions of these objects even in a small speck of the brain, up through seven dimensions. In some networks, we even found structures with up to 11 dimensions.”
Cliques of Neurons Bound into Cavities Provide a Missing Link between Structure and Function
- 1Blue Brain Project, École Polytechnique Fédérale de Lausanne, Geneva, Switzerland
- 2Laboratory for Topology and Neuroscience, Brain Mind Institute, École Polytechnique Fédérale de Lausanne, Lausanne, Switzerland
- 3Laboratory of Neural Microcircuitry, Brain Mind Institute, École Polytechnique Fédérale de Lausanne, Lausanne, Switzerland
- 4DataShape, INRIA Saclay, Palaiseau, France
- 5Institute of Mathematics, University of Aberdeen, Aberdeen, United Kingdom
Figure 1. (A) Thin (10 μm) slice of in silico reconstructed tissue. Red: A clique formed by five pyramidal cells in layer 5. (B1) Full connection matrix of a reconstructed microcircuit with 31,146 neurons. Neurons are sorted by cortical layer and morphological type within each layer. Pre-/postsynaptic neurons along the vertical/horizontal axis. Each grayscale pixel indicates the connections between two groups of 62 neurons each, ranging from white (no connections) to black (≥8% connected pairs). (B2) Zoom into the connectivity between two groups of 434 neurons each in layer 5, i.e., 7 by 7 pixels in (A), followed by a further zoom into the clique of 5 neurons shown in (A). Black indicates presence, and white absence of a connection. (B3) Zoom into the somata of the clique in (A) and representation of their connectivity as a directed graph.
2.1. The Case for Directed Simplices
Figure 2. (A1) A 4-clique in the undirected connectivity graph has one of 729 configurations in the directed graph. (A2) Configurations containing bidirectional connections are resolved by considering all sub-graphs without bidirectional connections. (A3) Without bidirectional connections, 64 possible configurations remain, 24 of which are acyclic, with a clear sink-source structure (directed simplices, in this case of dimension 3). (B) Number of simplices in each dimension in the Bio-M reconstruction (shaded area: standard deviation of seven statistical instantiations) and in three types of random control networks. (C) Examples of neurons forming high-dimensional simplices in the reconstruction. Bottom: Their representation as directed graphs. (D) (Left) Number of directed simplices of various dimensions found in 55 in vitro patch-clamp experiments sampling groups of pyramidal cells in layer 5. (Right) Number of simplices of various dimensions found in 100,000 in silico experiments mimicking the patch-clamp procedure of (B).
2.2. An Abundance of Directed Simplices
2.2.1. Reconstructed Neocortical Microcircuitry
2.2.2. Control Models
2.2.3. In vitro
2.2.4. C. elegans
2.2.5. Simplicial Architecture of Neocortical Microcircuitry
Figure 3. (A1) Number of simplices in each dimension in the excitatory subgraph (shaded area: standard deviation across seven instantiations). (A2) Same, for the inhibitory subgraph. (A3) Same, for the subgraphs of individual layers. (B) Distribution across seven instantiations of the Bio-M graph of the number of 3- simplices an excitatory (red) or inhibitory (blue) neuron belongs to (simplices/neuron). (C) Mean over neurons in individual layers of the highest dimension of a simplex that they belong to. (D) Simplices/neuron by layer and dimension. (E) Correlation of 3-simplices/neuron and degree in the graph for all neurons.
2.3. Topology Organizes Spike Correlations
Figure 4. (A) Patterns of thalamic innervation in the reconstruction. Each circle represents the center of innervation of a thalamic fiber. Each color represents a unique thalamic spike train assigned to that fiber. (B1) Exemplary directed simplex in a microcircuit. (B2) Connectivity and morphological types of neurons in the exemplary simplex. (B3) Raster plot and PSTH (Δt = 10 ms) of spiking response of neurons in (B1,B2) to stimulus S30b. (B4) Correlation coefficients of all pairs of PSTHs in (B3). (C) Correlation coefficients of PSTHs for all stimuli and all connected pairs of neurons in a microcircuit (Δt = 25 ms). (D) Mean correlation coefficients for connected pairs of neurons against the number of maximal simplices the edge between them belongs to, dimension by dimension. Means of fewer than 1,000 samples omitted. (E) Mean correlation coefficient of pairs of neurons, given their position within a simplex and its dimension.
2.4. Cliques of Neurons Bound into Cavities
Figure 5. (A) Example of the calculation of the Euler characteristic of a directed flag complex as an alternating sum of Betti numbers or simplex counts. (B) Euler characteristic against the highest non-zero Betti number (β5) for seven instances of reconstructed microcircuits based on five different biological datasets (Bio 1-5). (C) Top: The transmission-response (TR) graph of the activity of a microcircuit is a subgraph of its structural connectivity containing all nodes, but only a subset of the edges (connections). Bottom: An edge is contained if its presynaptic neuron spikes in a defined time bin and its postsynaptic neurons spikes within 10 ms of the presynaptic spike. (D) Fraction of edges active against fraction of high-dimensional simplices active in TR graphs for various time bins of a simulation. Error bars indicate the standard deviation over 10 repetitions of the simulation. Blue triangles: 4-dimensional simplices, blue squares: 5-dimensional simplices. Red symbols and dashed lines indicate the results for choosing edges randomly from the structural graph and the number expected for random choice, respectively.
2.5. Cliques and Cavities in Active Sub-Graphs
Figure 6. (A) Number of edges, β1, β3, and Euler characteristic of the time series of TR graphs in response to the stimulus patterns shown in Figure 4 (mean and SEM of 30 repetitions of each stimulus). (B) Trace of the time series of β1 against β3 for three of the stimuli. Shading of colors indicates Gaussian profiles at each time step with means and standard deviations interpolated from 30 repetitions of each stimulus. (C) Trace for one of the stimuli in B, along with the mean firing activity at different locations of the microcircuit during time steps of 2 ms. (D) Like (B), but for TR graphs of Bio 1-5, in response to stimulus S15b.
4. Materials and Methods
4.1. The Topological Toolbox
4.1.1. Directed Graphs
4.1.2. Simplices, Simplicial Complexes, and Flag Complexes
4.1.3. Simplicial Complexes of Directed Graphs
4.1.4. Directionality of Directed Graphs
18.104.22.168. Betti numbers
22.214.171.124. Euler characteristic
4.2. Computation of Simplices and Homology
4.2.1. Generating Directed Flag Complexes with Hasse Diagrams
4.2.2. Homology Computations
4.3. Model of Neocortical Microcircuitry
4.4. Control Networks
4.4.1. ER-Model (Random-Independent Graph)
4.4.2. PR-Model (Morphology-Only, “Peters' Rule”)
4.4.3. GB-Model (Shuffled, Preserving Distance Dependance)
4.5. Patch Clamp Experiments
4.5.1. In vitro
4.5.2. In silico
4.6. C. elegans Connectome
4.7. Simulation of Electrical Activity
4.7.1. Thalamic Stimulation
4.8. Spike Train Correlations
4.9. Transmission-Response Matrices
4.10. Data Analysis and Statistical Tests
Conflict of Interest Statement
Citation: Reimann MW, Nolte M, Scolamiero M, Turner K, Perin R, Chindemi G, Dłotko P, Levi R, Hess K and Markram H (2017) Cliques of Neurons Bound into Cavities Provide a Missing Link between Structure and Function. Front. Comput. Neurosci. 11:48. doi: 10.3389/fncom.2017.00048
Published: 12 June 2017.
Paul Miller, Brandeis University, United States
Cees van Leeuwen, KU Leuven, Belgium
Andreas Knoblauch, Hochschule Albstadt-Sigmaringen, Germany
*Correspondence: Henry Markram, email@example.com
Kathryn Hess, firstname.lastname@example.org
†These authors have contributed equally to this work.