Transcript A d

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
Neurons
Network functionality is captured by
neuronal connectivity
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
Anterior
Posterior
• Well documented
– Wiring diagram
– Neuronal map
1mm
• Simple system
– 302 neurons
– 11 gangalia
Nervous system
A
P
• One-dimensional
problem
Can wiring minimization predict
neuronal placement?
To the actual placement…
?
Pre-synaptic Neuron
Pre-synaptic Neuron
From the wiring diagram…
Post-synaptic Neuron
Chemical synapse
Electrical synapse
Post-synaptic Neuron
A
P
Quadratic Cost Function
1

2
2
E    Aij (ri  rj )    Bkl (rk  fl ) 

 2 i, j
  k ,l
Internal
wiring cost
External
constraints
For symmetrized A, rewrite into matrix form…
E  [r T ( DA  A)r ]  [r T DB r  2r T Bf  const ]
L
Laplacian of A
Optimal placement coordinates:
ri = position of neuron i
f l = position of sensor/effector l
Aij = neuron i to neuron j
connection matrix
Bkl = neuron k to sensor/effector l
connection matrix
DAij   ij  Aip
p
DB ij   ij  Bip
p
r   L  DB  Bf
1
Actual vs. Predicted Neuron Positions
Predicted Position
Predicted
Actual
r
al
eral
Ventral cord
Ventral
Lumbar
Pre-anal
Retrovesicular
Dorsorectal
Posterolateral
A
Actual Position
P
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
actual neurons
Synapse
Dendrites
Axons
Big brains - large numbers
Brain ~ 1011 neurons
Neuron ~ 104 synapses
Synapse
10cm
1mm
1mm
Assembling the wiring diagram will take many years
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
N
Number of neurons:
Wire diameter:
d
Axon length per neuron:
Total wiring volume:
R
 Network size:
Mouse cortical column (1mm3): N=105,
l
3
NR
Nld
2


R Nd
d=0.3mm 
 R=3cm 
Design II: Branching axons
(multi-pin nets)
R / N 1/3
Inter-neuron distance:
Axon length per neuron:
Total wiring volume:
 Network size: R
Cortical column:
R

l = R N 2/3
3
Nld
2

56
N d
N=105 d=0.3mm  R=4.4mm 
Design III: Branching
axons and dendrites
Total number of voxels: R 3 /
d3
Number of voxels containing axon:
Fraction of voxels containing axon: ld 2 /
R3
Fraction of voxels containing dendrite: ld 2 /
Number of voxels containing
axon and dendrite: l2d /R3 ~1
Total wiring volume:
R
3
Cortical column: N=105
Nld
2


l/d
R3
Network size:
R
23
N d
d=0.3;1mm R=1.6mm 
Is it possible to improve on Design III?
In Design III, dendrite length can be found…
R
3
R
Nld
23
2
N d

 l ~Nd
l
…to be smallest possible:
L>Nd
d
Design III cannot be improved if dendrites are
smooth
Design IV: Branching axons and
spiny dendrites
Number of voxels containing
axon and dendrite: l2s /R3 ~1
Total wiring volume:
R
 Network size: R
Cortical column: N=105
3
Nld
23
N d
2
43


13
s
d=0.3;1mm s=2.5mm 
 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
Mode-locked laser
IR
PMT
whiskers
Genetically engineered
mouse expresses GFP in
a small subset of neurons
Two-photon microscope provides in vivo images
with single-synapse resolution
Spine remodeling indicates synapse rearrangement in vivo
day 1
7
5
6
2
3
4
8
axon
dendrite
2mm
Trachtenberg, …, Svoboda, 2002
What determines axon (dendrite)
diameter?
t2
t1
d d d
t0
3
0
3
1
d0
Axon diameter minimizes the combined cost
of wiring volume and conduction delays
3
2
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
90% potential connectivity neighborhood
L1
L2
L3
L4
L5
100mm
Arbor reconstructions:
Hellwig, 2000
“Potential” definition of a cortical column
What is the correct cost function?
Biology: Min{V} -> Min{C=V–llogN}
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
Estimates of filling fraction from anatomical
data
Pyramidal
neuron density
n [105 mm3 ]
Dendritic
length/neuron
Ld [mm]
Interbouton
interval
b [ m m]
Spine
length
s [ m m]
Filling
fraction
f
Mouse
neocortical areas:
Mos, VISp
0.78
3.5
4.5
2.0
0.26
Rat hippocampal
areas:
CA3
0.21
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
V2
2.2*
1.3*
1.4
1.6
6.4
6.4
2.6*
2.1*
0.12
0.23
V4
1.1*
2.1
6.4
2.2*
7a
0.80*
2.6
6.4
2.1*
0.20
0.23
* Original data (Collaboration with Hof lab at Mount Sinai)
Neuronal morphology
Salient features:
• Axons
• Dendrites
• Branching
• Spines
Hof lab
What is the function of these features?
Number of potential synapses
2s
2s
Np - number of
potential synapses
s - spine length
La - axon length
Ld - dendrite length
n - neuron density
Ld
N p  2sLa Ld n
La
Number of potential synapses for
random orientation of axons
2s
Np - number of
Np 

2
potential synapses
sLa Ld n
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
Ad
Aa
Ld
La
Minimize V = LaAa + LdAd while NP ~ LaLd = const
Minimize LaAa + LdAd while NP ~ (LaAa) (Ld Ad) = const
Minimum V when LaAa = LdAd
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
C= 1 (t0  t1 )   2 (t0  t2 )   (V0  V1  V2 )
C  (1  2 )t0  V0   1t1  V1    2t2  V2 
13
 2(1   2 ) 
d0  

 k  
13
 21 
d1  

 k  
3
1
t0
13
 2 2 
d2  

 k  
d d d
3
0
t2
t1
3
2
d0
Why is axon-only wiring inefficient?
…
…
Long axons
Short dendrites
…
…
Short axons
Long dendrites
Dendrites enhance wiring efficiency
in highly convergent circuits
Job description of the nervous
system
Sensors
Nervous
system
Effectors
Cortical architecture is optimized for
high inter-connectivity
Synapse re-arrangement is potential memory mechanism with high
information storage capacity (Stepanyants, Hof, Chklovskii, 2002)