Blue Brain Team Discovers a Multi-Dimensional Universe in Brain Networks
Using mathematics in a novel way in neuroscience, the Blue Brain Project shows that the brain operates on many dimensions, not just the three dimensions that we are accustomed to.
For most people, it is a stretch of the imagination to understand the world in four dimensions but a new study has discovered structures in the brain with up to eleven dimensions – ground-breaking work that is beginning to reveal the brain’s deepest architectural secrets.
Using algebraic topology in a way that it has never been used before in neuroscience, a team from the Blue Brain Project has uncovered a universe of multi-dimensional geometrical structures and spaces within the networks of the brain.
The research, published today in Frontiers in Computational Neuroscience, shows that these structures arise when a group of neurons forms a clique: each neuron connects to every other neuron in the group in a very specific way that generates a precise geometric object. The more neurons there are in a clique, the higher the dimension of the geometric object.
Topology in neuroscience: The image attempts to illustrate something that can not be imaged – a universe of multi-dimensional structures and spaces. On the left is a digital copy of a part of the neocortex, the most evolved part of the brain. On the right are shapes of different sizes and geometries in an attempt to represent structures ranging from 1D to 7D and beyond. The “black-hole” in the middle is used to symbolise a complex x of multi-dimensional spaces, or cavities. Courtesy of the Blue Brain Project
“We found a world that we had never imagined,” says neuroscientist Henry Markram, director of Blue Brain Project and professor at the EPFL in Lausanne, Switzerland, and co-founder and Editor-in-Chief of Frontiers, “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 eleven dimensions.”
Markram suggests this may explain why it has been so hard to understand the brain. “The mathematics usually applied to study networks cannot detect the high-dimensional structures and spaces that we now see clearly.”
If 4D worlds stretch our imagination, worlds with 5, 6 or more dimensions are too complex for most of us to comprehend. This is where algebraic topology comes in: a branch of mathematics that can describe systems with any number of dimensions. The mathematicians who brought algebraic topology to the study of brain networks in the Blue Brain Project were Kathryn Hess from EPFL and Ran Levi from Aberdeen University.
“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,” explains Hess.
In 2015, Blue Brain published the first digital copy of a piece of the neocortex — the most evolved part of the brain and the seat of our sensations, actions, and consciousness. In this latest research, using algebraic topology, multiple tests were performed on the virtual brain tissue to show that the multi-dimensional brain structures discovered could never be produced by chance. Experiments were then performed on real brain tissue in the Blue Brain’s wet lab in Lausanne confirming that the earlier discoveries in the virtual tissue are biologically relevant and also suggesting that the brain constantly rewires during development to build a network with as many high-dimensional structures as possible.
When the researchers presented the virtual brain tissue with a stimulus, cliques of progressively higher dimensions assembled momentarily to enclose high-dimensional holes, that the researchers refer to as cavities. “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,” says Levi. “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.”
The big question these researchers are asking now is whether the intricacy of tasks we can perform depends on the complexity of the multi-dimensional “sandcastles” the brain can build. Neuroscience has also been struggling to find where the brain stores its memories. “They may be ‘hiding’ in high-dimensional cavities,” Markram speculates.
Original research article: Cliques of Neurons Bound into Cavities Provide a Missing Link between Structure and Function
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
This research was funded by: ETH Domain for the Blue Brain Project (BBP) and the Laboratory of Neural Microcircuitry (LNMC); The Blue Brain Project’s IBM BlueGene/Q system, BlueBrain IV, funded by ETH Board and hosted at the Swiss National Supercomputing Center (CSCS); NCCR Synapsy grant of the Swiss National Science Foundation; GUDHI project, supported by an Advanced Investigator Grant of the European Research Council and hosted by INRIA.
“Using mathematics in a novel way in neuroscience, the Blue Brain Project shows that the brain operates on many dimensions, not just the three dimensions that we are accustomed to.”
To be honest, it is not true. Tozzi and Peters demostrated in different papers (including Frontiers) that the brain is multidimensional. Differences: Markram demonstrated it in artificial microcolumns using novel variants of simplicial complexes, while Tozzi and Peters demonstrated it in the REAL whole human brain, using novel variants of the Borsuk-Ulam theorem.
Five friends have each other on speed-dial. One could graph this and come up with a pentagram, which is also the vertex map of the 4D polytope, the pentachoron. Does this mean that higher spatial dimensions are somehow involved in their cellphone chatter? I rather think not. But then again I am not seeking funds to research this underwhelming factlet, much less heralding a breakthrough.
Maybe the best reply to anything, ever.
The Frontiers in Computational Neuroscience rely on 22bit visual binary code, behaving like fiber optic cables relaying strings of data across electrochemical barriers and connections. Scientists around the world have united in the FriendshipCube project to introduce the 22bit visual binary code in neuroscience applications, bridging the divide between human and machine. I have successfully relayed 22bit electrical signals from my brain using the Emotiv EPOC EEG and “friendshipcube” software. Proving that my brain can send single bit signals in strings of 22bits or more. The visual binary cube code I use is graspable. We can hold it in our hands, and feel the symbols, and intuit their rotating meaning, as correspondences between light and sound, or geometry and phonetics, giving rise to the intuitive meanings of its 22 character alphabet. Somewhere between the Blue Brain Project and Friendship Cube Group, is a kind of Brain Trust, where the visual binary codes first uploaded to my personal computer will be indefinitely stored.
The brain works in many d-ment-ions. Who’da thunkit? What would we do w/t science?
I need a definition of ‘dimension’ as used here. Is it the same dimension as used in string theory or is the meaning specific to topology, of which I am fairly ignorant?
There are two different ways to define and assess brain dimensions. Indeed, the term dimension may reflect either
a) functional relationships of brain activities, or
b) anatomical connections between cortical areas.
a) The first approach takes into account the dimensionality of the neural space. Connectivity and complex network analyses of neural signals allow the assessment of the complex dynamics of brain activity, providing a novel insight into the multidimensionality of various neural functions’ representations (Kida et al., 2016). From a dynamical system perspective, one would expect that brain activities are represented as, for example, some scalar quantity measured at different brain locations (say N locations) at different points in time. Then one could describe nervous dynamics as trajectories and/or manifolds in a N-dimensional phase space (Lech et al., 2016). Mazzucato et al (2016) demonstrated that stimuli reduce the dimensionality of cortical activity. Clustered networks, such as default mode network, have instead a larger dimensionality, because the latter grows with ensemble size: the more neurons are recruited, the more the dimensions (Mazzucato et al, 2016). Apart from giving insights in neural dynamics in the canonical three dimensions (space, time, and frequency), complex network analyses are also able to evaluate other functional dimensions, e.g. categories of neuronal indices such activity magnitude, connectivity, network properties and so on (Kida et al., 2016). It must be taken into account that dimension reduction and symmetry breaking display close relationships, so that symmetries are correlated with changes in functional dimensions in the brain. Indeed, a key feature of dynamical approaches is that the dynamics they predict are characterized by nonequilibrium phase transitions, and therefore breaks of symmetries (Scholz et al., 1987). Many studies emphasized how different levels of behavioral dynamics’ organization take place in neural ensembles. To make some examples, Jirsa et al. (1998), focusing on the cortical left-right symmetry, derived a bimodal description of the brain activity that is connected to behavioral dynamics, while Jirsa et al. (1994) demonstrated that, when an acoustic stimulus frequency is changed systematically, a spontaneous transition in coordination occurs at a critical frequency, in both motor behavior and brain signals.
b) Concerning the second approach to brain dimensionality, it has been recently suggested that brain trajectories, at least during spontaneous activity, might display four spatial dimensions, instead of three (Tozzi and Peters 2016). Brain symmetric states display dimensions higher than asymmetric ones, so that, in this case, the space of interest does not refer to dynamical neural spaces, but to detectable physical cortical locations. In such a vein, Stemmler et al. (2015) proposed that animals can navigate by reading out a simple population vector of grid cell activity across multiple spatial scales. Combining population vectors at different microscopic dimensions predicts indeed neural and behavioral correlates of multiscale grid cell readout, that transcend the known link between entorhinal grid cells and hippocampal place cells. While the spatial activity of a single grid cell does not constitute a metric, an ensemble of hierarchically organized grid cells does provide instead a distance measure (Stemmler et al., 2015). In our paper, the mapping of trajectories from high dimensional manifold to lower dimensions refers to both the above described definitions of dimensionality.
In sum, the study of changes in brain dimensions is a promising novel methodology. We need to take into account that, despite neural networks modelling complex systems are known to exhibit rich, lower-order connectivity patterns at the level of individual nodes and edges, however higher-order organization remains largely unknown. Benson et al. (2016) recently developed an algorithmic framework for studying how complex networks are organized by higher-order connectivity patterns, revealing unexpected hubs and geographical elements. In such a vein, Kleinberg et al. (2016) demonstrated that real networks are not just random combinations of single networks, but are instead organized in specific ways dictated by hidden geometric correlation between layers. Such correlations allowed the detection of multidimensional communities, e.g., sets of nodes that are simultaneously similar in multiple layers. Crucial for our topological arguments, such multidimensionality also enables accurate trans-layer link prediction, meaning that connections in one layer can be predicted by observing the hidden geometric space of another layer. For example, when the geometric correlations are sufficiently strong, a multidimensional framework outperforms navigation in the single layers, allowing efficient targeted navigation simply by using local multilayer knowledge (Kleineberg et al., 2016).
Maybe Stanislaw Lem was on the right track in “Solaris”. Sand Castles rising up and fading away on the surface?