Brain Criticality - Optimizing Neural Computations

Summary

The video script delves into the critical brain hypothesis, which likens the brain's neural networks to a system at the brink of phase transition, akin to Maxwell's equations for electromagnetism. It explains phase transitions, critical points, and their relevance to neuroscience, suggesting that the brain operates near a critical point to optimize information processing. The script explores concepts like neuronal avalanches, power laws, and the balance between excitation and inhibition, highlighting the brain's computational efficiency at the critical state.

Takeaways

• 🧠 Understanding the brain's complexity is a major challenge in neuroscience, with a goal to develop a comprehensive theory similar to Maxwell's equations for electromagnetism.
• 🌟 The critical brain hypothesis has gained attention; it suggests that neural networks operate near a phase transition point, akin to the critical state of water molecules coexisting in liquid and gas phases.
• 🔍 The concept of phase transitions is introduced with the example of water boiling, illustrating how energy can lead to a change in the macroscopic properties of a system.
• 📊 Phase transitions are characterized by an order parameter and driven by a control parameter, with first-order transitions showing a discontinuity and second-order transitions being continuous.
• 🌡 The Ising model, used to explain magnet properties, is highlighted as an example of a system undergoing a second-order phase transition, leading to a critical point with unique properties.
• 🌐 At the critical point, systems exhibit long-range communication and scale-free behavior, with self-similar patterns observable at all scales, a characteristic of fractals.
• 📉 Neuronal avalanches, observed in neural cultures, are bursts of activity that spread across networks and are indicative of the brain operating near a critical point.
• 📈 The distribution of features of neuronal avalanches follows a power law, suggesting a lack of characteristic scale in brain activity and supporting the idea of criticality in neural systems.
• 🔄 The balance between excitation and inhibition in the brain is crucial and can be adjusted to demonstrate phase transitions, with compounds that block these processes causing shifts from critical dynamics.
• 💡 Operating near a critical point is beneficial for the brain as it optimizes information processing and computational power, allowing for efficient transmission and reduction of uncertainty about inputs.
• 📚 The study of criticality in neuroscience is an exciting and growing field, with potential applications in understanding brain function and developing treatments for neurological disorders.

Q & A

What is one of the Holy Grails of Neuroscience?

One of the Holy Grails of Neuroscience is to build an elegant yet comprehensive theory that describes the general roles of the nervous system, similar to the impact of Maxwell's equations on electromagnetism.

What does the critical brain hypothesis propose about the operation of networks of neurons?

The critical brain hypothesis proposes that networks of neurons operate near a point of phase transition, in a special critical state similar to when water molecules coexist in liquid and gaseous phases.

How does the phase transition concept relate to the boiling of water?

The phase transition concept relates to the boiling of water as an example of a first-order transition, where water changes from a liquid to a gaseous state, absorbing latent heat and undergoing a qualitative change in properties.

What is an order parameter in the context of phase transitions?

An order parameter is a macroscopic property that quantifies the organization of a system, such as entropy, volume, fluidity, or surface tension in the case of water.

How does the control parameter influence the phase transition?

The control parameter, such as temperature or pressure, drives the phase transition by altering the system's state. It is an independent variable that can be freely varied in an experiment.

What is a continuous or second-order phase transition?

A continuous or second-order phase transition is a type of phase transition where the change in the order parameter is continuous, allowing the system to smoothly transition from one phase to another.

What is the significance of the critical point in a phase transition?

The critical point is a unique intermediate state at the interface of two different phases where the boundaries are blurred, and new properties emerge. It is a point of balance between order and disorder.

How does the Ising model relate to the concept of criticality in the brain?

The Ising model, originally developed to explain the properties of magnets, demonstrates how a system can undergo a second-order phase transition, exhibiting properties of criticality such as long-range communication and scale-free behavior, which are relevant to understanding neural networks.

What is the role of the correlation length in the Ising model?

The correlation length in the Ising model is the distance at which the dynamic correlation between spins drops to zero. It indicates the extent of long-range communication in the system, peaking at the critical point.

What is the significance of scale-free behavior in the context of the brain?

Scale-free behavior signifies that the system, such as the brain, has no characteristic scale and resembles itself at any scale. This property is indicative of criticality and is associated with optimal information processing and computational power.

How does the branching ratio (Sigma) act as a control parameter in neural networks?

The branching ratio (Sigma) is a control parameter that governs the transition from subcritical to supercritical dynamics in neural networks. When Sigma equals 1, the network is at a critical point where activity is neither decaying nor amplifying, and information transfer is optimized.

What experimental evidence supports the idea that the brain operates near a critical point?

Experimental observations such as neuronal avalanches, which are power-law distributed patterns of activity in neural networks, support the idea that the brain operates near a critical point. These have been observed in various species and scales, from single neurons to EEG recordings.

Why is operating near a critical point considered beneficial for the brain?

Operating near a critical point is considered beneficial for the brain because it maximizes capabilities such as information processing and computational power. At the critical point, the brain can efficiently transmit information and maintain a balance between activity and quiescence.

Outlines

00:00

🧠 The Quest for a Neural Theory and Critical Brain Hypothesis

This paragraph introduces the complexity of understanding the brain and the pursuit of a comprehensive theory in neuroscience, akin to Maxwell's equations for electromagnetism. It presents the critical brain hypothesis, which suggests that neural networks operate near a phase transition point, drawing an analogy to the coexistence of water in liquid and gaseous states. The paragraph sets the stage for exploring the concept of criticality and phase transitions in the context of neural information processing.

05:00

🌡 Understanding Phase Transitions and Critical Points

The paragraph delves into the concept of phase transitions, using the example of water boiling to illustrate the qualitative change from liquid to gas. It explains order parameters and control parameters in phase transitions and introduces first and second order transitions. The critical point, where the system can exist in an intermediate state with unique properties, is highlighted as a point of interest for understanding the brain's operation.

10:03

🧲 The Ising Model and Its Relevance to Neuroscience

This section introduces the Ising model, originally developed for magnets, as a tool for understanding neural networks. It describes the model's lattice structure, where each site can be in one of two states, and how the alignment of these states can lead to macroscopic properties like magnetization. The paragraph explains how temperature acts as a control parameter, inducing fluctuations that can prevent the system from settling into the most energetically favorable configuration, and discusses the model's continuous second-order phase transition.

15:05

🔗 Long-Range Communication and Correlation Length

The paragraph explores how long-range communication can emerge from local interactions in the Ising model, introducing the concept of correlation length to measure the coordinated behavior of distant sites. It discusses how dynamic correlation changes with temperature and how the system becomes scale-free at the critical point, with self-similar patterns observable at different scales.

20:06

📊 Power Laws and Scale Invariance in Critical Phenomena

This section discusses the power law distribution observed in the Ising model at the critical point, which indicates scale invariance. It uses a thought experiment to illustrate how the probability distribution of cluster sizes follows a power law, leading to the mathematical definition of scale invariance. The importance of power laws in understanding critical phenomena and phase transitions is emphasized.

25:07

🌐 Neuronal Avalanches and the Brain's Critical State

The paragraph presents experimental evidence suggesting that the brain operates near a critical point, with neuronal avalanches observed in neural cultures. It describes how these avalanches resemble activity cascades in other systems and how they are characterized by power law distributions. The paragraph also discusses the potential implications of criticality for neural information processing across species and scales.

30:07

🔄 The Branching Model and Neural Information Transmission

This section introduces the branching model to represent neural activity, where each neuron is connected to others with a transmission probability. It explains how the branching ratio, a control parameter, governs the system's behavior, leading to a phase transition at a critical value. The paragraph discusses how this model can be used to understand the balance between excitation and inhibition in the brain and the emergence of scale-invariant spike avalanches.

35:08

🎯 Optimizing Information Processing at the Critical Point

The paragraph explores why operating near a critical point is beneficial for the brain, maximizing its information processing capabilities. It uses a guessing game analogy to illustrate how the critical point optimizes information transfer, with the output layer of the branching model resembling the input layer's activity. The importance of this balance for neural computation and the potential applications of criticality research in neuroscience are highlighted.

🚀 Conclusion and the Future of Criticality Research in Neuroscience

In conclusion, the video summarizes the key properties of systems undergoing second-order phase transitions, such as scale-free behavior and long-distance communication. It emphasizes the role of the critical point in optimizing the brain's information processing and the potential of criticality research for understanding brain function and developing treatments for neurological disorders. The video also promotes further exploration and discovery in this exciting field.

Keywords

💡Neuroscience

Neuroscience is the scientific study of the nervous system, which includes the brain, spinal cord, and nerves. It is central to understanding how neurons and neural networks process information. In the video, neuroscience is the overarching field of study that seeks to build comprehensive theories, like the critical brain hypothesis, to describe the general roles and operations of the nervous system.

💡Critical Brain Hypothesis

The critical brain hypothesis is a theory suggesting that networks of neurons operate near a point of phase transition, akin to a critical state in physical systems. It has gained attention due to accumulating experimental evidence. The hypothesis is integral to the video's exploration of how the brain may function optimally at the edge of a phase transition, balancing between order and disorder.

💡Phase Transition

A phase transition refers to a change in the state of matter, such as from liquid to gas, driven by changes in control parameters like temperature or pressure. In the context of the video, the brain's phase transition is metaphorical, suggesting a shift in the state of neural activity. The script discusses how the brain might undergo a phase transition in terms of neural activity rather than physical state changes.

💡Order Parameter

An order parameter is a macroscopic property that quantifies the organization of a system. In the video, it is used to characterize the phase of a system, such as entropy in the case of water. The concept is applied to the brain's neural networks, where the order parameter might represent the level of neural activity or connectivity.

💡Control Parameter

A control parameter is an independent variable that can be adjusted to induce phase transitions in a system. Examples include temperature or pressure. In the context of the brain, the script suggests that the balance between excitation and inhibition acts as a control parameter, influencing whether neural networks are subcritical, critical, or supercritical.

💡Second-Order Transition

A second-order transition is a type of phase transition where the order parameter changes continuously. The video highlights second-order transitions as they relate to the brain's critical state, where the system can exhibit scale-free behavior and long-range communication without a characteristic scale.

💡Icing Model

The Ising model is a mathematical model of ferromagnetism in statistical mechanics, which the video uses as an analogy to explain critical phenomena in neural networks. The model demonstrates how local interactions can lead to long-range correlations and scale-free behavior at critical points.

💡Correlation Length

Correlation length is a measure of the distance over which correlations in a system decay. In the video, it is used to describe how far information can propagate in a system at criticality. The script mentions that at the critical point, correlation length extends to a larger distance, indicating effective communication across the network.

💡Scale Invariance

Scale invariance refers to the property of a system where its statistical properties remain invariant under changes in scale. The video explains that at the critical point, neural networks become scale-free, with no characteristic scale, and exhibit self-similar patterns at all levels of magnification.

💡Power Law

A power law is a functional relationship between two quantities, where a relative change in one quantity results in a proportional relative change in the other quantity, independent of the initial size. In the video, power law distributions are observed in the sizes and durations of neuronal avalanches, suggesting the brain operates near a critical point.

💡Neuronal Avalanches

Neuronal avalanches refer to the cascades of neural activity that propagate through a network, resembling avalanches in physical systems. The video discusses how these avalanches, characterized by power-law distributions, indicate that the brain may be operating at a critical state, optimizing information processing.

💡Branching Ratio

The branching ratio, denoted as Sigma in the video, is a parameter that represents the average number of neurons activated by a single neuron. It acts as a control parameter for neural networks, with a value of 1 indicating a critical state where activity is neither amplified nor diminished, optimizing information transfer.

Highlights

Understanding the brain's complexity is a major challenge in neuroscience, with the goal of developing a comprehensive theory similar to Maxwell's equations for electromagnetism.

The critical brain hypothesis has gained attention, suggesting that neural networks operate near a phase transition point, akin to water's liquid and gaseous states coexistence.

The concept of phase transitions, such as water turning into gas, is used to explain the critical state in neural networks without actual melting or boiling.

Criticality in the brain is related to the efficient processing of neural information, with the phase transition being a significant point of interest.

Order parameters and control parameters are key in characterizing phase transitions, with the brain's critical point being driven by the balance of excitation and inhibition.

Second-order phase transitions, unlike first-order, involve continuous changes in the order parameter, allowing for the existence of a critical point with unique properties.

The Ising model, used to explain magnet properties, is also applied to understand critical points in neural networks, demonstrating long-range communication from local interactions.

Correlation length and dynamic correlation are measures used to understand how information is transmitted across neural networks at criticality.

Scale-free systems, like those at critical points, exhibit self-similarity at all scales and are characterized by power law distributions.

Neuronal avalanches, observed in neural cultures, are activity patterns that spread across networks and are indicative of critical brain dynamics.

Power law distributions of neuronal avalanches suggest the brain operates near a second-order phase transition, with no characteristic scale.

The branching model simplifies the complexity of neural networks, illustrating how activity can propagate and lead to critical phenomena.

The branching ratio, Sigma, is identified as a control parameter for neural networks, with a value of 1 indicating criticality and optimal information transfer.

Operating near a critical point maximizes the brain's information processing capabilities and computational power.

Criticality in neuroscience has potential applications in understanding brain function and developing treatments for neurological disorders.

The book 'The Cortex and the Critical Point' by John Max provides an in-depth look at criticality in neuroscience.

Brilliant.org is highlighted as an educational platform for learning computational methods and analysis of complex data sets like those in neuroscience.

Transcripts

understanding the brain in all of its

complexity is a difficult challenge one

of the Holy Grails of Neuroscience is to

build an elegant yet comprehensive

Theory describing the general roles of

the nervous system

similar to what a set of Maxwell

equations was to electromagnetism

in this video we are going to talk about

the critical brain hypothesis

a theory which has been getting a lot of

attention and experimental evidence in

recent years

it states that networks of neurons

operate near a point of phase transition

it is special critical State similar to

when water molecules coexist in liquid

and gaseous phases but what does it even

mean for the brain to undergo a phase

transition

there are surely no melting and boiling

right and in what way is this critical

point whatever it is important to neural

information processing all of that and

more coming right up

to understand the concept of criticality

let's begin with the notion of phase

Transitions and one such transition that

is probably very intuitive to you is

when liquid turns into gas consider a

pot of water on a hot stove at first

when the water is below 100 degrees the

energy provided by the burning stove

Heats it up accelerating individual

molecules at 100 degrees however

something interesting happens

even though the heat is still being

pumped into the water

temperature stays constant instead of

accelerating individual water molecules

the energy is now being spent on

Breaking the bonds between them

which allows the molecules to break free

from the lattice and fly away as a gas

notice that at the boiling point we have

seen a qualitative switch from liquid

which is incompressible has surface

tension and can dissolve substances to

gas which is compressible and doesn't

really dissolve anything even though the

individual water molecules stayed the

same the macroscopic properties of the

system as a whole changed drastically

what we have just witnessed is known as

the phase transition

more generally a phase transition is

observed when a system moves from

existing in one well-defined state of

organization or phase to another

phase is usually characterized by an

order parameter a macroscopic property

that somehow quantifies the organization

of the system for example in the case of

water order parameter can be entropy the

degree of disorder volume fluidity or

surface tension the phase transition is

driven by changes in control parameter

for example temperature or pressure

essentially you can think of the control

parameter as an independent variable

something that we can freely vary in an

experiment and the order parameter is a

dependent variable which changes in

response to altering the control

parameter and quantifies the state of

our system

notice that if you plot the order

parameter as a function of control

parameter then in the case of water

boiling you'll see a discontinuity a

sudden jump as we go from liquid to gas

accompanied by the absorbance of latent

energy such phase transitions are called

discontinuous or first order transitions

these are what you normally see in

everyday life however we will be more

interested in another type so called

continuous or second order transitions

as you might have guessed from the name

in this type of transitions the change

in the order parameter is continuous

which means that you can smoothly go

from one phase to another

now the slope of this curve can be very

steep sure but it's always continuous

this allows the system to exist in a

unique intermediate State called the

critical point right at the interface

when the boundaries between the two

different phases are blurred and new

properties emerge this is what makes the

critical point so special as we'll see

further

water can actually undergo a second

order transition at specific values of

temperature and pressure this is where

latent heat disappears and water can be

continuously transformed into gas

existing in a special state known as

supercritical fluid but to develop

intuition about the properties of

critical point let's focus on a

different much simpler system where the

second order transition feels more

natural and which actually has a lot of

applications in neuroscience

meet the icing model which was first

developed to explain the properties of

magnets the model consists of a large

lattice where each site can be in one of

two states plus one or -1 these sites

represent individual particles that the

system is made of characterized by their

spins which you can think about as this

sign of magnetic field generated by each

particle

when all spins are aligned tiny magnetic

fields of individual particles add up

resulting in the magnetic properties on

a macro scale

however when these pins are pointing in

different directions individual magnetic

fields essentially cancel each other out

and the system has no macro scale

magnetization

physicists have known for a long time

that if you heat up a magnet past a

certain point called Curie temperature

it will suddenly lose its magnetic

properties so there must be some sort of

phase transition at play now there are

two types of interactions that govern

the Dynamics of the system

first of all neighboring spins will tend

to line up because it is more

energetically favorable

we can write the energy of a single

pairwise interaction between the two

sides has been proportional to the

negative product of their spins where

the coefficient of proportionality J is

called the coupling constant and it

tells us how strongly the sides are

coupled as you can see when the spins

are of the same signs the energy of an

interaction is negative while for a pair

of opposite spins the value of energy is

positive

to find the energy of one site all we

need to do is to sum the energies of the

interactions between its four neighbors

and summing together the energies for

all these Heights will give us the

energy of the entire system

because it will try to minimize its

energy we can expect all these pins to

perfectly line up

so what can prevent this from happening

notice that so far we haven't included

our control parameter temperature in any

of the equations in reality however

flipping of spins is a stochastic

process which is subject to random

thermal movement

more formally the distribution of energy

values subject to random fluctuations

obeys the boltzmann distribution which

gets more broad at higher temperatures

but it's not that important to

understand phase transitions so don't

worry about it

essentially you can think of temperature

as this stochastic destabilizing force

that will induce fluctuations Jocelyn

spins around and preventing them from

settling down into their most

energetically favorable configuration

the higher the temperature the more

disordered the system would be

to quantify this let's introduce an

order parameter for example the

macroscopic magnetization which is

defined as the average value of spins

when the absolute value of magnetization

is big

it means that a large number of spins

are aligned and our system will display

magnetic properties let's play with the

temperature and see how macroscopic

properties of the system change at low

temperatures local interactions Dominate

and the system exists in this state when

a large portion of spins are lined up in

contrast if you heat it up the

temperature related fluctuations take

over all the spins are oriented randomly

and no macroscopic magnetic properties

are observed notice that as we turn the

temperature knob there is something that

looks like a very abrupt change in the

organization of spins from Euler to

disorder

indeed it turns out that icing model

undergoes a continuous second order

phase transition

in other words it is possible to catch

the system in an intermediate state

right at the point of critical

temperature where order and disorder are

perfectly matched and a number of

interesting properties emerge

before we move to the brain let's spend

a little more time exploring the icing

model

similarly to how neurons communicate

with each other across the entire brain

there is a crosstalk between spins in

our lattice

notice that if you flip once pin this

can make some of the neighboring spins

to flip which in turn affects pins two

sides away and so forth So eventually

even though only nearest neighbor

interactions are explicitly defined the

information about flipping one spin has

the potential to be transmitted across

the entire ladders

in other words long-range communication

emerges from purely local interactions

a good way to measure this is through a

quantity called correlation length for

example let's take two lattice sites

some distance apart and look at their

Dynamic correlation which is a measure

of how coordinated in time their

behavior is

more formally for sites I and J we can

express the dynamic correlation as

follows where angular brackets indicate

averaging over time the first term in

the parenthesis represents the amount by

which the spin I fluctuates from its

average value and likewise for

fluctuations of the spin J

notice that in order to maximize the

value of dynamic correlation the spins

should fluctuate in a coordinated

fashion for the product of the

parenthesis to remain positive let's see

what happens to the value of dynamic

correlation for a given pair as we

change the control parameter

at low temperatures the spins don't

fluctuate much so they are stuck at

their average values yielding a low

value of dynamic correlation

at high temperatures the spins fluctuate

wildly but they do so in a random

uncoordinated fashion at one point in

time the terms in the parenthesis might

be of the same sign while on another

occasion they might be of different

signs so on average the value of dynamic

correlation is low as well

at the critical temperature however

nearest neighbor interactions are

balanced by the thermal stochasticity

there is a coexistence of coordination

and fluctuation resulting in the large

value of dynamic correlation as spins

flip up and down in a coordinated way

so far we've only looked at the dynamic

correlation for a single pair of pins

and that of course will depend on the

distance between them let's now plot the

value of c as a function of distance

between the spins for different

temperatures

unsurprisingly it decreases with

distance because we've only Incorporated

local interactions into the model

and what's prominent is that the curve

corresponding to the critical

temperature decays much more slowly than

the two other ones for example we might

find that it extends up to 20 sites

before it falls to near zero let's turn

this distance at which the dynamic

correlation drops to zero as correlation

length

drawing correlation length as a function

of temperature reveals a sharp Peak at

the critical point right there at the

intersection of odor and disorder is

where a large distance communication

emerges if we were to make an analogy

this is the state in which neurons in

networks communicate most effectively

through the largest number of synapses

between them

another important property that emerges

at the critical point is that our system

becomes scale free to illustrate what

this means take a look at these three

magnified snapshots of the isig model at

three different temperatures exactly

critical slightly below and slightly

above the critical temperature can you

guess which is which well this is not so

obvious because they all look kind of

similar but let's zoom out to see the

whole lattice as we move further and

further away it becomes apparent that

the leftmost snapshot corresponds to a

sub-critical temperature because

everything is ordered

zooming out on the rightmost snapshot

reveals a great deal of disorder so this

must be this super critical State now

watch what happens as we zoom out when

the system is at the edge of the phase

transition no matter how close or how

far away we are things look very similar

in the ideal case right at the critical

point this pattern will continue to

Infinity in other words there is no

characteristic scale this

self-similarity when the system as a

whole resembles some of its parts at any

scale is a typical property of fractals

such zooming animations are surely

satisfying to look at but is there any

way we can objectively prove that this

is scale free while this is not

to mathematically describe the scaling

properties let's consider the

probability distribution of this size of

clusters that is connected domains of

the same spin

here are the examples of a few

relatively small clusters and a couple

of larger ones let's focus on the

critical case and build the intuition

for what this function f should look

like

suppose a dark wizard has shrunken you

to a random size so that you have no

idea how tall you are and trapped you in

the icing model at the critical

temperature the only way for you to get

out is to correctly guess the

probability distribution of cluster

sizes

so you grab a ruler and start walking

around measuring the size of every

cluster you see

let's say on average you find out that

every hundredths cluster you encounter

has the size of four square inches so

you can write down F of 4 equals 1 over

100 because the ruler has been shrunken

with you the exact units are irrelevant

what would matter in this case are

ratios so you begin looking for clusters

that are two times as large and discover

that the probability of observing an 8

square inches cluster is about 1 over

500 which is 5 times as large compared

to the probability of seeing a four

square inches one after repeating the

measurements for a few other cluster

sizes you discover an interesting

pattern every time you double the size

of a cluster its probability decreases

by a factor of five

now here is the crucial Point Let's

imagine that you were shrunken to a

completely different size for example

what if you were actually 10 times

taller what would you see in this case

well remember because the system is

right at the critical point it is scale

invariant

in other words no matter how small or

big you were you would see similar

pictures everywhere and hence similar

cluster distributions at any scale so

this ratio of the probability for

observing the cluster of size 2x versus

the cluster of size X would equal to

one-fifth at every scale

which means it doesn't depend on x

of course there is nothing special about

the factor of 2. we could have just as

well looked at any other ratio of

cluster sizes so let's substitute the

number 2 with a factor k

the value of this ratio will be also

different for different factors because

for instance the probability of

observing a cluster with a size 3x will

be different than the probability of

seeing a cluster of size 2x in general

we can write down the right hand side to

be some function J of K since as we have

already established it surely doesn't

depend on x

this right there is the mathematical

definition of scale invariance so

whatever the distribution of cluster

sizes is it should satisfy this

constraint it turns out that the only

function for which this equation holds

is of the form a times x to the power of

minus gamma where the number gamma is

called the exponent in that case J of K

is equal to K to the power of minus

gamma plugging in our values for cluster

sizes reveals that gamma equals 2.32

this behavior is called a power law and

it is central for the study of critical

phenomena you can easily see for

yourself that x to the power of minus

gamma is indeed scale free proving it

the other way around that any scale free

function should have this form is a bit

more tricky and requires taking a few

derivatives I won't go through the

derivation in this video but if you are

interested check out the references in

the description

another way to see why power laws are

scale free is to plot a power law

function for example x to the power of

minus 2.3 next to something similar but

which is not power law for example an

exponential function notice that if we

look at the graphs at different scales

the shape of the parallel curve stays

the same the only thing that changes are

the exact units if axis labels were

erased we couldn't distinguish them from

each other however the exponential curve

looks different at different scales and

hence is not scale free

usually such relations are plotted in

double logarithmic coordinates known as

log lock plots

where you take the logarithm of both

sides in log lock coordinates power laws

look like straight lines and the slope

of that line is related to the exponent

value gamma

importantly many other characteristics

of this system for example the value of

dynamic correlation we discussed earlier

all follow a power law distribution near

a critical point in fact when you see

that a lot of things are power law

distributed it often suggests that the

system is near a point of a second order

phase transition

now after we've developed an intuition

for many important Concepts in

criticality let's finally move to the

brain

[Music]

one of the first pieces of experimental

observations suggesting that the brain

might be operating near a critical point

came in 2003 when John Banks ended our

plans published their seminal paper

titled vironal avalanches in neocortical

circuits they grew cultures of neurons

taken from Red somatosensory cortex and

put them on an 8x8 grid of electrodes to

record spontaneous activity that arises

in this network and what they found was

a characteristic pattern of activity

spreading across the network in space

and time let's take a closer look at the

data collected by a single electrode it

records how extracellular voltage known

as local field potential changes with

time

if you have seen my video on theater

rhythms it is essentially the same thing

what is usually observed in these

cultures of neurons in a dish however is

that Baseline lfp occasionally shows

large deflections following a

characteristic shape of a negative Spike

superimposed on a positive envelope this

arises when a number of neurons in the

vicinity of electrode activate

simultaneously

we are going to treat such waveforms as

discrete events that are either present

or absent simply by finding when the

voltage Trace dips below a certain

threshold and also keep track of the

amplitude of every such event

thus our data will contain both a Time

point and an amplitude value for every

lfp event that caused the threshold

repeating this procedure for all the

electrodes gives us the pattern of

spontaneous electrical activity

spreading in the network

notice how periods of quiescence are

interspersed by what looks like chains

of activity arising reverberating in the

network and finally dying out

such bouts of activity were termed

neuronal Avalanches because they

resemble Cascades of activity

propagating in systems like pile of sand

or earthquakes

to Define an avalanche more formally

let's break up the entire time course

into short means and for each bin count

the number of electrodes that recorded

the lfp event for example it might go

like this

first there is no activity then one

electrode detects the Supra threshold

event

in the next time being another electrode

then two simultaneously then another one

then three others at the same time and

then activity dies out

this sequence of time bins with some

non-zero activity flanked by at least

one time being of quiescence at either

side is what we are going to call an

avalanche

and we can describe each Avalanche by

its duration as well as its size defined

as the number of electrodes that were

activated during the Cascade

note that in the alternative definition

size can also take into account the

amplitude values

surprisingly the authors found that all

these features of neuronal avalanches

were distributed according to a power

law revealing a prominent straight line

in log lock plots

this suggested that the network activity

had no characteristic scale and that the

brain might be operating near a point of

a second order phase transition

now all of this happened in 2003 and

since then Pablo described Avalanches of

activity were demonstrated in awake

behaving animals of many species

including worms zebrafish monkeys and

humans and on a variety of scales from

single neuron recordings to

electroencephalography

suggesting that criticality can indeed

be a universal phenomenon which

describes many aspects in neural systems

at this point in the video I think a

couple of pieces are still missing first

of all I've said that the critical point

is the intermediate stage of phase

transition but what are the distinct

phases for the case of the brain what

would be neural analogues to temperature

as the control parameter and

magnetization as the order parameter and

secondly why operating near a critical

point would be useful to the brain at

all

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to make the following discussion more

intuitive let's switch from thinking

about avalanches in terms of lfp events

on the electrode grid and view the data

as activity of individual neurons that

either fire an action potential or not

such switch is Justified because of

scale invariants remember we are going

to see similar Behavior at any scale

plus it has been established

experimentally that there are indeed

such Spike Avalanches whose features are

power law distributed

in this representation each circle is an

individual neuron rather than a single

electrode now it becomes easier to

interpret the activity propagation in

the network

whenever any run reaches its threshold

voltage it sends a signal to its

Downstream Partners increasing or

decreasing their probability of

subsequent firing

but because neurons connect to each

other in very intricate ways unlike the

icing model for example it is hard to

see who is connected to whom

let's make a slight simplification to

this network and rearrange the neurons

into a layered structure so that

information will always flow from left

to right

even though we don't allow any feedback

connections or Loops to a first

approximation it is a reasonable

assumption and importantly this

description known as branching model

still allows for avalanches and other

critical phenomena to emerge in the

branching model each neuron is randomly

connected to some neurons in the

downstream layer and each connection has

a transmission probability associated

with it

upon this Spike of a sender neuron the

connection will transmit a spike with a

certain probability and if this happens

on the next time point the receiver

neuron will activate and send the

information further

additionally each neuron has a very

small probability of spontaneously

activating even when it doesn't receive

any input this stochasticity provides

the network with some input drive to

work with

as you may have guessed how activities

spreads in the network will be mostly

controlled by the distribution of

transmission probabilities to tune this

with a single parameter let's introduce

a number Sigma called a branching ratio

and set the sum of our going

transmission probabilities for each

neuron to be equal to Sigma

for example if the branch in ratio is 1

this configuration of our going

connections is valid because the sum of

probabilities is equal to 1 and this is

not

by the way the branching ratio doesn't

really tell us the exact number of

connections for Sigma equal to 1 a

neuron can have a hundred connections

each with a probability of 100 or one

connection with the probability of one

the only thing that is constrained is

the overall sum of transmission

probabilities

it turns out that this branching ratio

is a control parameter for our system

and that a phase transition occurs when

Sigma is equal to 1.

to understand why this is the case

notice from the very definition of Sigma

it follows that it is actually equal to

the average number of descendants that

are activated per single active ancestor

neuron

let's run the simulation for different

values of branching ratio and see what

happens

as you can see when Sigma is equal to

0.5 any activity that rises quickly dies

out this is because on average it takes

two Upstream neurons to fire in order to

activate a single Downstream neuron

in a real brain that would correspond to

deep comatose state with very little

activity

increasing the value of Sigma 2 causes

the network to blow up with activity

similar to what you would see during

epilepsy this is because on average

single neuron activates two others which

makes the activity amplify with time

something interesting happens when Sigma

is equal to 1 however

in this configuration one neuron on

average activates one Downstream

descendant allowing the network activity

to be maintained on a relatively

constant level without decaying or

amplifying

now because this is a stochastic process

Avalanches will eventually die out but

their sizes and durations will be power

law distributed

this is why the branch in ratio is a

control parameter governing the

transition from decaying to amplifying

activity and we can use the average

density of active neurons as the order

parameter which is really low for

subcritical values of branches ratio but

quickly grows as soon as we hit the

critical point by the way in real

neurons this control parameter which is

equal to the average number of neurons

activated by a single Upstream neuron is

shaped by the balance between excitation

and inhibition which are the two

counteracting forces in the brain

Dynamics importantly because this

balance of excitation and inhibition is

something that we can control

experimentally it can further illustrate

that there is indeed a phase transition

at play namely adding compounds that

block inhibitory transmission

disrupted the power law distribution of

avalanches seen in the normal condition

leading to supercritical behavior

likewise adding compounds that block

excitatory synaptic transmission leads

to subcritical Dynamics again with the

disruption of power laws

okay by now the only left is why would

brain even evolve to operate near a

critical point is it actually useful it

turns out that at the critical point a

lot of brain's capabilities such as

information processing and computational

power are maximized to begin

understanding why this is the case let's

think of information transmission as a

guessing game

suppose someone provides an input to the

first layer of our branching model by

activating a random set of neurons and

our job is to correctly guess the number

of neurons that were activated by

observing only the output the activity

of the rightmost layer

for low values of branching ratio no

matter how many neurons were activated

in the beginning the activity quickly

vanishes before reaching the output

layer we are completely ignorant about

the input because there is no activity

Trace left

for high values of Sigma activity will

be Amplified causing the output layer to

be fully activated even for the weakest

input which makes the guessing again

very difficult

at the critical case when Sigma is equal

to 1 the connection strengths are tuned

to the optimal intermediate value

because one neuron on average activates

only one descendant activity of the

output layer has a high probability of

resembling the activity of the input

layer in terms of the number of active

units

in other words observing the output

successfully reduces our uncertainty

about the input

if we introduce some measure of

information transmission that will tell

us how accurately we can make this gas

this quantity will have a sharp Peak

when the branch in ratio is 1.

similar to how Dynamic correlation Peaks

at the critical temperature in the icing

model this on a very simplified level is

what it means to optimize information

transfer

of course we have really just scratched

the surface of the topic of critical

point and its relevance to Neuroscience

I didn't really talk about the concept

of universality and the Beautiful

relationship between the exponents the

idea of course a criticality and how the

brain actually maintains this right

balance in the first place but if you

are interested to know more about

criticality and face transitions in the

Neuroscience context I really encourage

you to check out a wonderful book

published in 2022 titled the cortex and

the critical point by John Max who is

one of the pioneers of this field the

book is written in a really accessible

language and introduces all the concepts

from the ground up

before we move to the summary I have one

important message

in this video we have seen a few

examples of how analyzing neural data

can help us better understand the brain

but have you ever wondered how computers

can be programmed to process and analyze

such complex data sets if you're

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alright let's recap in this video we

have explore

X properties that arise when the system

undergoes a second order phase

transition namely it becomes scale free

and there is an emergence of long

distance communication between the

components

in the brain such phase transition is

governed by the fine balance between

excitation and inhibition and by

hovering near a critical point our

brains optimize information processing

of course this area is still very new so

there is a lot of things we don't know

but it is now apparent that the

applications of criticality research are

far-reaching and exciting from

understanding the general principles of

brain function to developing new

treatments and therapies for

neurological disorders as our

understanding of criticality in neural

networks continues to grow it is sure to

be an exciting field of exploration and

discovering in the years to come

if you like the video share it with your

friends and colleagues subscribe to the

channel if you haven't already press

like button and consider supporting me

on patreon stay tuned for more

interesting topics coming up

goodbye and thank you for the interest

in the brain foreign

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