inside any brain there is a certain pattern of connections that keeps showing up everywhere whether you look at individual neurons in a Petri dis or connectivity between major brain areas and humans there is something Universal in all of them and you know what's even more intriguing when sociologists analyze friendship networks or when molecular biologists look at how genes regulate each other they all find the exact same pattern it's as if nature keeps converging on this this particular way of organizing complex networks today we're taking a closer look at this mysterious pattern of organization called small
world graphs we'll discuss what they are why they appear everywhere you look and how they give your brain such remarkable information processing capabilities before we begin let's establish some terminology at their core networks are collections of elements called nodes connected by Links called edges together nodes and edges form a graph an abstract representation of the network structure this abstraction is powerful precisely because nodes and edges can represent many different physical systems in the brain for instance nodes might be individual neurons with edges representing synaptic connections between them or we might zoom out treating entire brain
regions as nodes with edges corresponding to white matter tracts connected them the beauty is that these fundamentally different systems can be analyzed using the same mathematical framework once we abstract them as graphs even more Connections in the graph don't have to be physical at all Beyond anatomical connectivity that is the physical wiring of neurons or brain regions we can study functional connectivity which captures how different parts of the system work together two neurons might be considered functionally connected if tend to generate electrical impulses in synchrony or two brain regions might show coordinated activity during a
specific task while functional connections often have a lot in common with the underlying anatomical paths they reveal a different layer of organization showing us not just how the brain is wired but how that wiring gives rise to coordinated activity everything we'll cover in this video was demonstrated both within various anatomical and functional networks revealing Universal principles of network organization at all scales and modalities now that we have these basic concepts at hand let's explore what makes certain graph architectures particularly special imagine you're designing a system like a city a computer network or a brain that
needs to perform two competing functions on one hand you need tight-knit local groups where elements work closely together in a city these are neighborhoods where people frequently interact in the brain these are local circuits processing specific information like recognizing edges in an image or controlling hand movements on the other hand you also need efficient Global Communication sometimes information needs to quickly travel between distant parts of the system like a delivery from one side of City to another or a signal from neurons in a visual area to motor pathways now is there any way you can
achieve both well the simplest solution would be to connect everything to everything else but that is incredibly costly and in the physical world often Impossible Too Many Roads too many cables or in the brain's case too many long range connections taking up space volume inside the skull is finite so there is only so many Pathways the brain can afford while it might look like these two objectives are in direct competition with each other it turns out there is an optimal solution that that efficiently combines The Best of Both Worlds local specialization and Global integration and
to understand it we need to look at the two fundamental properties of graphs the efficiency of Global Communication in a network can be Quantified by its average path length if you randomly select any pair of nodes what is the minimum number of edges you need to Traverse to get from one to another this measure tells us how well connected the network is globally how efficiently information can flow between any two points a low average path length indicates that any part of the network can quickly communicate with any other part local interaction density can be formalized
through what's known as the clustering coefficient let's take any node X and consider its immediate neighbors the nodes it connects to directly for these neighbors we examine how interconnected they are with each other in networks with high clustering we expect these neighbors to share many connections among themselves think of human social interactions if you have two friends chances are they're also friends with each other through things like work or school to quantify this mathematically we can compute the ratio of actual connections between all the neighbors of X and the total number of possible connections between
them for example if the node has three neighbors there are three possible edges that could exist between those neighbors if all of these possible connections are present the clustering coefficient for this node X is one the maximum possible value if only one of these potential connections exists the clustering would be 1/3 to characterize the network as a whole we can calculate the clustering coefficient for each node and take their average which tells us how prevailing tight need group groups are throughout the system with those two quantities at hand let's see how different network architectures balance
them first consider a regular lattice like the arrangement of atoms in a crystal where each node connects only to its nearest Neighbors in such networks cherine is naturally High your neighbors tend to be neighbors with each other however to get from one side of the network to another you need to Traverse many many steps making Global Communication highly inefficient with large path lengths at The Other Extreme imagine connecting nodes completely randomly even with just a small fraction of possible edges such networks achieve remarkably short path lengths allowing you to reach any point in just a
few hops however they lack meaningful local structure for networks with a limited number of connections clustering coefficient is very low because your neighbors are no more likely to connect to each other than to any other random node but is there some kind of Sweet Spot between those two extremes this very question led to a breakthrough in network science it turns out you can dramatically reduce path length while maintaining most of the clustering by rewiring just this small fraction of Connections in a regular lattice this random process of rewiring certain local connections is known as the
wats stoats model adding just a few shortcuts a longrange connections that bridge distant parts of the network is enough to make the graph globally efficient while preserving its local structure networks that achieve this balance High clustering almost like a regular Lis but short path lengths like a random graph are called small world networks alluding to the phenomenon when you meet a stranger with surprising Mutual connections and exclaim what a small world speaking of unexpected connections they appear not just between people but also between ideas in knowledge networks these conceptual small worlds are exactly what our
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the what strats model is a specific simple method for creating random graphs with such Small World Properties but not all small world networks including the brain can be described using this simple model it's hard to imagine that neurons in the brain start off as a regular lattice that is randomly rewired indeed there are certain other properties of brain networks Beyond being just small world that are not captured by this model and perhaps the most important of them is the existence of hubs when we look at real brain networks we find something striking not all nodes
are created equal some neurons or brain regions have many more connections than others if we plot the distribution of how many connections different nodes have known as their degree we don't see the Bel shaped curve you'd expect from a simple wat strats model where all nodes have similar connection patterns instead we find what's known as a heavy tailed distribution often close to Lo normal or power lock curves this means that there is a small but significant number of Highly connected nodes that play a special role acting as kind of hubs in the network these hubs
serve as Bridges between different processing modules enabling efficient Global Communication while preserving specialized local processing we see this principle at work across many different scales in SE Elegance warms where scientists have mapped every single neuron and every connection there indeed exist certain Hub cells responsible for relaying a bulk of communication within the network in the human brain we find specific regions that send connections throughout much of the brain one such Hub is the locus serui a nucleus in the brain stem that acts as the brain's primary source of noradrenaline modulating arousal and attention across diverse
brain regions this architectural principle combining small world networks namely short paths and high clustering with specialized Hub nodes appears consistently across multiple scales in the brain its repeated emergence suggests that it's a fundamental principle of neural organization but why did evolution Converge on this particular arrangement we initially arrived at small world architecture as graphs that solve a fundamental mathematical problem namely achieving both short path lengths and high clustering coefficients with reasonable number of connections well it turns out this Maps perfectly to the challenges faced by evolving nervous systems at any given moment different parts of
your brain are processing specific types of information with high degree of parallelism in your visual cortex some groups of neurons detect edges others track motion While others process color these specialized groups need to be able to work independently like parallel computers each focused on their own task extracting many different features at once without having to wait for each other this is where High clustering of small world networks becomes crucial it creates modules of neurons that can work together intensively without interference from other groups and we see this specialized module processing throughout the brain your somar
sensory cortex for example has distinct patches of neural tissue dedicated to processing touch Sensations from different body parts motor areas have separate circuits controlling different muscle groups each of these modules needs its own space to process information efficiently but special ized computation alone isn't enough the brain needs to rapidly combine information from these different modules to create coherent experiences and actions imagine catching a ball your visual system needs to communicate with motor areas to guide hand movements you need to integrate information about the ball's speed trajectory and position with precise muscle control all in a
fraction of a second this requires Global efficiency the ability to quickly transmit signals between any two points in the network this is where the magic of small world architecture truly shines these crucial shortcut connections and especially hob neurons dramatically reduce the number of steps needed for signals to travel between distant brain regions information can Traverse millions of neurons in just a few synaptic hops of course this Global connectivity comes at a cost as each longrange connection require Ires energy to maintain and takes up valuable space inside the skull the longer the A on wire the
more cellular Machinery is needed to transport proteins and maintain the connection this wiring cost problem is exactly why the brain can't simply connect everything to everything it needs to achieve maximum computational power with minimum possible wiring another fascinating benefit of the small world connectivity is its robustness to failure in individual neurons are not particularly reliable after all they are biological cells that can die or malfunction Evolution has pushed our brains to be resilient to such disruptions the dense connectivity within modules creates redundancy if one component fails information can find alternative Roots while random damage to
single neurons typically has minimal impact there are certainly points of vulnerability those crucial Hub nodes damage to these highly connected nodes can have widespread effects throughout the network which may help explain why certain brain disorders can cause such diverse symptoms when key hob regions are affected all right let's tie everything together in this video we explored small world networks and elegant graph architecture that combines short path lengths with high clustering this network structure appears throughout nature from Gene networks to social systems in the brain we find it at every scale suggesting that these patterns are
fundamental to computation indeed small world networks solve three crucial challenges for the brain they enable specialized processing in local circuits allow rapid integration of information across distant regions and maintain robustness against failure it is a brilliant compromise between the competing demands of efficiency and cost showing us how Nature has discovered fundamental principles of network design that are Universal in so many domains of life if you enjoyed the video share it with your friends subscribe to the channel if you haven't already and press the like button stay tuned for more neuroscience and machine learning topics coming
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