Evolution as the blind engineer: wiring minimization in the brain

Dmitri “Mitya” Chklovskii

Cold Spring Harbor Laboratory

Optimization is a powerful theoretical tool for understanding brain design

• Evolutionary theory: survival of the fittest

• Maximize fitness to predict animal design

• Fitness ~ functionality – cost

• Minimize cost for given functionality

Brain as a neuronal network

Sensors Effectors

Network functionality is captured by neuronal connectivity

Neurons

Evolutionary cost of wiring

• Signal delay and attenuation

• Metabolic requirements

• Space constraints

• Guidance defects in development

Wiring cost grows with the distance between connected neurons

For given functionality minimize wiring length

C. elegans as Model System

• Well documented– Wiring diagram– Neuronal map

• Simple system– 302 neurons– 11 gangalia

• One-dimensional problem

Anterior Posterior

A P

Nervous system

1mm

Chemical synapse

Electrical synapse

Can wiring minimization predict neuronal placement?

?

From the wiring diagram… To the actual placement…

A P

Post-synaptic Neuron

Pre

-syn

aptic

Neu

ron

Post-synaptic Neuron

Pre

-syn

aptic

Neu

ron

Quadratic Cost Function

2 2

, ,

1( ) ( )

2 ij i j kl k li j k l

E A r r B r f

ir = position of neuron i

ijA = neuron i to neuron j connection matrix

klB = neuron k to sensor/effector l connection matrix

lf = position of sensor/effector l

Internal wiring cost

External constraints

For symmetrized A, rewrite into matrix form…

[ ( ) ] [ 2 ]T T TA BE r D A r r D r r Bf const

A ij ipijp

D A

B ij ipijp

D B L

Laplacian of A

1

Br L D Bf Optimal placement coordinates:

Actual vs. Predicted Neuron Positions

Actual

AnteriorDorsal

LateralVentral

RetrovesicularPosterolateral

Ventral cordPre-anal

DorsorectalLumbar

Predicted

A P

Actual Position

Pre

dict

ed

Po

sitio

n

Wiring minimization is reasonable but not perfect

Why is not wiring minimization prediction perfect?

• Nervous system may be sub-optimal

• Other constraints may be important

(e.g. development)

• Quadratic cost function may be incorrect

• Routing optimization may affect placement

Routing or neuronal shape

point neurons

Axons

Dendrites

Synapseactual neurons

Big brains - large numbers

10cm

Brain ~ 1011 neurons

Assembling the wiring diagram will take many years

Neuron ~ 104 synapses

1mm

m

Synapse

Routing problem

• Network of N neurons

• Fully connected (all-to-all)

• Fixed wire diameter, d

Find wiring design minimizing network volume

Design I: Point-to-point axons

l NRAxon length per neuron:

3 2R NldTotal wiring volume:

Number of neurons: N

Mouse cortical column (1mm3): N=105, d=m

R NdNetwork size:

R=3cm

Wire diameter: d

Design II: Branching axons (multi-pin nets)

3 2R NldTotal wiring volume:

Inter-neuron distance: R / N 1/3

Cortical column: N=105 d=m R=4.4mm

Network size:5 6R N d

Axon length per neuron: l = R N 2/3

Design III: Branching axons and dendrites

3 2R NldTotal wiring volume:

Number of voxels containing axon: l/d

Cortical column: N=105 d=m R=1.6mm

Total number of voxels: R 3 / d 3

Fraction of voxels containing axon: ld 2 / R 3

Fraction of voxels containing dendrite: ld 2 / R 3

Number of voxels containing

axon and dendrite: l2d /R3 ~1Network size:

2 3R N d

Is it possible to improve on Design III?

l

dDesign III cannot be improved if dendrites are

smooth

2 3R N d

3 2R Nld l ~Nd

In Design III, dendrite length can be found…

…to be smallest possible:

L>Nd

Design IV: Branching axons and spiny dendrites

3 2R NldTotal wiring volume:

Number of voxels containing

axon and dendrite: l2s /R3 ~1

2 3 4 3 1 3R N d sNetwork size:

Cortical column: N=105 d=m s=2.5m

R=0.8mm

Network volume for various wiring designs

Neuronal shape is a routing solution implementing high inter-connectivity

Cortical architecture is optimized for high inter-connectivity

Synapse re-arrangement is potential memory mechanism with high information storage capacity (Stepanyants, Hof, Chklovskii, 2002)

Experiments on synapse re-arrangement

Two-photon microscope provides in vivo images with single-synapse resolution

IR

PMT

Mode-locked laser

Genetically engineered mouse expresses GFP in a small subset of neurons

whiskers

axon

dendrite

day 1 axon

dendrite

day 2 axon

dendrite

day 3 axon

dendrite

day 4 axon

dendrite

day 5 axon

dendrite

day 6 axon

dendrite

day 7 axon

dendrite

day 8

Trachtenberg, …, Svoboda, 2002

Spine remodeling indicates synapse re-arrangement in vivo

2m

What determines axon (dendrite) diameter?

Axon diameter minimizes the combined cost of wiring volume and conduction delays

3 3 30 1 2d d d

d 1 d2

d0

t0

t1 t2

Summary

Wiring minimization is a key factor determining brain architecture

Complexity of neuronal networks poses challenging wiring minimization problems

Potential synapse is a location where axon comes within a spine length of a dendrites

• Potential synapse is a necessary (but not sufficient) condition for an actual synapse

• Potential synaptic connectivity is more stable than actual

• Potential synaptic connectivity can be evaluated geometrically

s

L1

L2

L3

L4

L5

90% potential connectivity neighborhood

Arbor reconstructions:Hellwig, 2000

“Potential” definition of a cortical column

100m

What is the correct cost function?

Biology: Min{V} -> Min{C=V–logN}

Physics: Min{E} -> Min{F=E–TS}

Constrained optimization is a powerful tool for building a theory of brain function

Acknowledgments

Armen Stepanyants

Cold Spring Harbor Laboratory

Interbouton interval

Pyramidal neuron density

Filling fraction

Estimates of filling fraction from anatomical data

5 3[10 ]n mm [ ]b m [ ]s m f

  Dendritic length/neuron

Spine length

Mouse neocortical areas:

Mos, VISp0.78 3.5 4.5 2.0 0.26

Rat hippocampal areas:

CA30.21 12.3 7.0 1.8 0.18

CA1 (CA3→CA1 projections)

0.47 10.8 3.0 1.8 0.23

Layer III of the Macaque monkey neocortical areas:

V1 2.2* 1.4 6.4 2.6* 0.12V2 1.3* 1.6 6.4 2.1* 0.23

V4 1.1* 2.1 6.4 2.2* 0.20

7a 0.80* 2.6 6.4 2.1* 0.23

[ ]dL mm

* Original data (Collaboration with Hof lab at Mount Sinai)

Neuronal morphology

Salient features:• Axons• Dendrites• Branching• Spines

What is the function of these features?

Hof lab

Number of potential synapses

Np - number of

potential synapses s - spine length

La - axon length Ld - dendrite length

n - neuron density

2p a dN sL L n

2s

Ld

2s

La

Number of potential synapses for random orientation of axons

2p a dN sL L n

2sNp - number of

potential synapses s - spine length

La - axon length Ld - dendrite length

n - neuron density

Equipartition of volume between axons and dendrites

Minimize total volume V for fixed NP and cross-sectional areas Aa , Ad

Minimize V = LaAa + LdAd while NP ~ LaLd = const

Minimize LaAa + LdAd while NP ~ (LaAa) (Ld Ad) = const

Minimum V when LaAa = LdAd

LaLd

AdAa

Large numbers of neurons & synapses and wide range of spatial scales

make the connectivity problem difficult to solve experimentally

but, at the same time, treatable with theoretical analysis!

Theoretical analysis

• Explains much of neuronal shape

• Can help infer connectivity from shape

• Predicts a potential memory mechanism

• Re-defines the connectivity problem

Optimal branch diameters

1 0 1 2 0 2 0 1 2( ) ( ) ( )t t t t V V V C=

1 2 0 0 1 1 1 2 2 2( )t V t V t V C

1 3 1 3 1 3

1 2 1 20 1 2

2( ) 2 2 d d d

k k k

3 3 30 1 2d d d

d 1 d2

d0

t0

t1 t2

Why is axon-only wiring inefficient?

… …

Long axonsShort dendrites

Short axons Long dendrites

……

Dendrites enhance wiring efficiency in highly convergent circuits

Job description of the nervous system

Sensors EffectorsNervoussystem

Cortical architecture is optimized for high inter-connectivity

Synapse re-arrangement is potential memory mechanism with high information storage capacity (Stepanyants, Hof, Chklovskii, 2002)