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 SensorsEffectors 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 AnteriorPosterior A P Nervous system 1mm

Chemical synapse Electrical synapse Can wiring minimization predict neuronal placement? ? From the wiring diagram…To the actual placement… AP Post-synaptic Neuron Pre-synaptic Neuron Post-synaptic Neuron Pre-synaptic Neuron

Quadratic Cost Function = position of neuron i = neuron i to neuron j connection matrix = neuron k to sensor/effector l connection matrix = position of sensor/effector l Internal wiring cost External constraints For symmetrized A, rewrite into matrix form… L Laplacian of A Optimal placement coordinates:

Actual vs. Predicted Neuron Positions Actual Anterior Dorsal Lateral Ventral Retrovesicular Posterolateral Ventral cord Pre-anal Dorsorectal Lumbar Predicted A P Actual Position Predicted Position 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 Synapse actual neurons

Big brains - large numbers 10cm Brain ~ neurons Assembling the wiring diagram will take many years Neuron ~ 10 4 synapses 1mm m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 Axon length per neuron: Total wiring volume: Number of neurons: N Mouse cortical column (1mm 3 ): N=10 5, d=  m Network size: R=3cm  Wire diameter: d

Design II: Branching axons (multi-pin nets) Total wiring volume: Inter-neuron distance: R / N 1/3 Cortical column: N=10 5 d=  m R=4.4mm  Network size: Axon length per neuron: l = R N 2/3

Design III: Branching axons and dendrites Total wiring volume: Number of voxels containing axon: l/d Cortical column: N=10 5 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: l 2 d /R 3 ~1 Network size:

Is it possible to improve on Design III? l d Design III cannot be improved if dendrites are smooth l ~Nd In Design III, dendrite length can be found… …to be smallest possible: L>Nd

Design IV: Branching axons and spiny dendrites Total wiring volume: Number of voxels containing axon and dendrite: l 2 s /R 3 ~1 Network size: Cortical column: N=10 5 d=  m s=2.5  m 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 2m2m

What determines axon (dendrite) diameter? Axon diameter minimizes the combined cost of wiring volume and conduction delays d1d1 d2d2 d0d0 t0t0 t1t1 t2t2

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 100  m

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 Dendritic length/neuron Spine length Mouse neocortical areas: Mos, VISp Rat hippocampal areas: CA CA1 (CA3→CA1 projections) Layer III of the Macaque monkey neocortical areas: V12.2* * 0.12 V21.3* * 0.23 V41.1* * a0.80* * 0.23 * 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 N p - number of potential synapses s - spine length L a - axon length L d - dendrite length n - neuron density 2s LdLd LaLa

Number of potential synapses for random orientation of axons 2s N p - number of potential synapses s - spine length L a - axon length L d - dendrite length n - neuron density

Equipartition of volume between axons and dendrites Minimize total volume V for fixed N P and cross-sectional areas A a, A d Minimize V = L a A a + L d A d while N P ~ L a L d = const Minimize L a A a + L d A d while N P ~ (L a A a ) (L d A d ) = const Minimum V when L a A a = L d A d LaLa LdLd AdAd AaAa

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 d1d1 d2d2 d0d0 t0t0 t1t1 t2t2

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 Effectors Nervous system

Cortical architecture is optimized for high inter-connectivity Synapse re-arrangement is potential memory mechanism with high information storage capacity (Stepanyants, Hof, Chklovskii, 2002)