Modelling and measuring maps of nerve connections

Part A

David WillshawInstitute for Adaptive & Neural Computation

School of InformaticsUniversity of Edinburgh

willshaw@inf.ed.ac.uk

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The talk has three parts

Review of my modelling work on the development of retinocollicular maps

• Spin off: How to measure the precision of a somatotopic map (Work in progress)

Spin off: Examination of the development of the olfactory nervous system

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PART 1:

Retinotectal Mapping Revisited:New data and old theories

Tadpoles

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Frog visual system

FROG BRAIN

CAT BRAIN

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What is the mechanism for the formation of ordered maps of nerve connections?

Probably more than one:1. involving electrical activity (neighbour matching by correlated activity?)2. involving molecular recognition (chemoaffinity?)

From Jacobson (1967)

Assorted evidence to be accounted for that indicates:

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Flexibility of connection pattern – usually associated with neural activity based mechanisms

Rigidity of connection pattern – usually associated with molecular based mechanisms

Regeneration studies in adult goldfish

- molecular recognition involving distinct immutable labels?

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Visuotectal map in stage 50-51 Xenopus

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Map in stage 55 Xenopus

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Visuotectal map in postmetamorphic Xenopus

– spreading of connections to fill available space?

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Evidence from surgically constructed compound eyesGaze, Straznicky, 1970s- 1980s - JEEM-> Development)

Continuous map from mirror symmetric compound-eye

Discontinuous maps from point-symmetric compound-eyes

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Mechanism 1: correlated neural activity

According to the correlated neural activity model, spontaneous electrical activity drives the process.

By a Hebbian-type mechanism, connections between neighbouring retinal cells and neighbouring tectal cells are strengthened; those between non-neighbours are weakened (Willshaw and von der Malsburg 1976).

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Mechanism 2, molecular recognition:Sperry’s chemoaffinity hypothesis (1943)

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Sperry’s chemoaffinity hypothesis (1943)

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Sperry’s chemoaffinity hypothesis (1943)

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Modern evidence for molecular labels

from McLaughlin, Hindges and O’Leary (2003)

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But are these molecules used in map-making?

If Ephs and ephrins are the labels of chemoaffinity, then changing them should result in abnormal patterns of connectivity

I have analysed the abnormal retinocollicular maps in mouse after knockin and knockout of specific EphA receptors in the retina

Mouse – retinocollicular projection

• The EphA3 receptor is knocked in to 50% of the retinal ganglion cells, dispersed over the entire retina (Brown et al., Cell (2000), 102, 77-88) In the map, the retina is projected twice over the colliculus, once over the front and once over the backThe EphA3+ cells form one projection, the normal cells the other

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Distribution of EphA receptors across the normal retina (Reber, Burrola & Lemke, Nature, October 2004)

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Reber et al. (2004) made 6 combinations of EphA3 knockin with EphA4 knockout

The distributions of total EphA density over the nasotemporal axis of the retina for each of the 6 cases:

Black: for EphA3- cells

Green: for EphA3+ cells

• Reber et al., Nature 2004 combined EphA3 knockin with EphA4 knockout, in each case obtaining a similar pattern of maps

I showed that in all these maps, the distribution of EphA receptor making up the gradient of EphA across the retina and present in the axonal terminals is distributed across the colliculus as a gradient from rostral to caudal

• Regardless of the absolute values forming the gradient

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The retinal induction model (Willshaw, 2006; after Willshaw and von der Malsburg, 1979)

provides a mechanism for this

A Law of Retinocollicular Map-Making: For the map of nasotemporal retina on rostrocaudal

colliculus, axons arrange themselves on the colliculus according to the relative amount of EphA they carry:

Temporal axons (highest EphA density) go to rostral colliculus

Nasal axons (lowest EphA density) go to caudal colliculus

Axons with intermediate amounts of EphA go to intermediate positions

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Rolling back the clock 30 years: we knew then that if map-making employs label matching,

then label flexibility is required

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Our solution back then: the Marker Induction model

(Willshaw and von der Malsburg, 1979)

• The retina carries fixed labelsThe retina carries fixed labels• A very rough map is formed A very rough map is formed

initiallyinitially

• Molecules are induced into the Molecules are induced into the tectum through each synapse tectum through each synapse formed, in amounts formed, in amounts characteristic of the axon’s cell characteristic of the axon’s cell of originof origin

• Exchange of molecules Exchange of molecules between tectal cellsbetween tectal cells

• Modification of synapsesModification of synapses

• Simultaneous refinement of Simultaneous refinement of tectal molecular distributiontectal molecular distribution

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The Marker Induction model

Initially there are labels in the retina and very weak labels, or none, in the target.

During development, simultaneously the set of fixed retinal labels is copied onto the target AND an ordered projection develops.

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According to the marker induction idea, chemical labels are induced onto the target structure from the retina

through the synaptic contacts made

The pattern of labels across the retina is duplicated in the target

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Reber et al. (2004) made 6 combinations of EphA3 knockin with EphA4 knockout

The distributions of total EphA density over the nasotemporal axis of the retina for each of the 6 cases:

Black: for EphA3- cells

Green: for EphA3+ cells

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How marker induction produces the double maps seen by Brown et al, 2000

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In conclusion of part 1The prevailing view of axons are guided initially by molecular signals to form a very rough map and then the map is formed by correlated neural activity.

• However, according to a 30 year old model, a mechanism of correlated neural activity is not required.

Axons can sort themselves out according to the amount of label they carry.

Many predictions: eg, the distribution of labels in colliculus will be different in each of the six knockins/outs discussed.

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PART 2

Spin off: measuring maps

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Read all about this in:

Willshaw (2006). Development, 133, 2705-2717

Willshaw & von der Malsburg (1979). Phil Trans Roy Soc B, 287, 203-243

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Since more than one mechanism is involved in map making, testing out the effect of one mechanism (eg, by genetic or surgical interference) will produce disordered maps

A method is needed to measure the order in such maps and hence assess the role played by the mechanism disrupted

The basic idea is to treat local and global order separately

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Since more than one mechanism is involved in map making, testing out the effect of one mechanism (eg, by genetic or surgical interference) will produce disordered maps

A method is needed to measure the order in such maps and hence assess the role played by the mechanism disrupted

The basic idea is to treat local and global order separately

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My goal is to provide quantitative measures of the local and global order in any map

I deal here with maps defined by the projection of a given number of points from one space onto the other

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My method is based on the fact that in an ordered mapping, neighbourhood relations are preserved

Establish which of the members of the set of points in one of the spaces are neighbours – say, the points in the tectum/colliculus

• Find the precision in which these neighbours map onto points in the other space (here, visual field).

• Find how individual parts of the map are oriented and in what overall orientation

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1) Definition of neighbours in tectum

Given a set of points distributed over the tectum:Use a Delaunay triangulation to define nearest neighbours.

23 points spread over the tectum Lines join nearest neighbours

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Given the set of points in the visual field corresponding to the given set of tectal points:•Join up each pair of field points for which the related tectal points are nearest neighbours•For each corresponding pair of field and tectal links, calculate the ratio of their lengths (after scaling map areas)

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2) Quantification of Map Precision

• In an ideal map, the lattice formed in the tectum will be reproduced exactly in the visual field, and so each ratio of link lengths will all be 1

• A single peak in the distribution of length ratios indicates homogeneity

• The standard deviation measures the orderliness of the map

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Counting the number of links that cross

• In a completely ordered map, the tectal lattice is reproduced perfectly in the visual field, with no links crossing

• In a disordered map, the number of links in the projected graph that cross over each other is a direct measure of (dis)order

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A partially disordered map – adult Xenopus

23 points, 53 crossings

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Distribution of NN link lengths:

23 points (disordered)

Mean: 1.56

Std Dev: 0.96

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3) Quantification of Map OrientationCompare the orientations of each of the sides of the matching Delaunay triangles

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Quantification of Map OrientationLook at the distribution of all such angles