Structural and functional properties of a nervous system:

Modelling tadpole locomotor behaviour in response to sensory signals

Roman Borisyuk, Andrea FerrarioUniversity of Exeter, UK

Robert Merrison-Hort City Research, Exeter UK

In collaboration with neurobiological laboratories of

Alan Roberts, Steve Soffe (University of Bristol)

Wenchang Li (University of St. Andrews)

Outline

• Introduction: structure and function of Neural Network (NN)

• Hatchling Xenopus tadpole is a unique animal to study structure and function of NN

• Developmental approach: axon grows and pair-wise connectivity

• Probabilistic model: generalisation from anatomical modelling

• Biologically realistic modelling of the tadpole nervous system

• Conclusions

Introduction• Information processing in the brain is based on

communication between spiking neurons that are embedded in a network of connections (current dogma).

• A resulting NN (Neuronal Circuit) is a traditional object for mathematical/computational modelling.

https://deskarati.com/2011/12/19/new-wonder-drug-could-give-us-all-super-memory/

Introduction• Information processing in the brain is based on

communication between spiking neurons that are embedded in a network of connections (current dogma).

• A resulting NN (Neuronal Circuit) is a traditional object for mathematical/computational modelling.

https://deskarati.com/2011/12/19/new-wonder-drug-could-give-us-all-super-memory/

Introduction

To design a Neural Network (NN) model the following three key characteristics have to be specified:

• Description of unit’s dynamics

• Connectivity (interactions) between units

• Learning rule (adjustment of connection strength) – we do not consider learning in our model

After that, the dynamics of neural activity can be simulated and activity patterns can be investigated.

From mathematical point of view, the NN activity is a solution of a large system of ODEs (or DDEs or stochastic DDEs).

Unit (Neuron) Activity: Action Potential (Spike)

Hodgkin-Huxley model (1952, Nobel Prize)

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Action Potential

There are two major connection types:

Electrical coupling (gap junction)Chemical synaptic connection

http://www.ncbi.nlm.nih.gov/books/NBK11164/

Connections and spiking

From modelling point of view, there are two major types of synaptic connections: excitatory and inhibitory connections. It means that a probability of action potential increases ordecreases respectively.

However, the neurobiology is much more complicated than this simple modelling scheme. For example, the Post-Inhibitory Rebound (PIR) mechanism provides a possibility to generate an action potential after inhibition:

Action potential Response to a short excitatory current injection and threshold property

Post-Inhibitory Rebound: Spike is generated after inhibitorycurrent injection

A large number of connections

Connections between units (connectome) is the most difficult part of NN specification.

• Usually, the number of units (N) is large and the number of connections grows as N2. Therefore, finding the connection architecture is a complex experimental problem.

• Theoretically, standard approaches of dimensionality reduction (e.g. from statistical physics) are not applicable because the neurons and their interactions are heterogeneous. There are many different types of neurons with specific properties for each cell type.

• Also, synaptic transmission is a very complex machinery with multiple interactive stochastic processes and components.

Variability of connectomes • It is known that brain development involves multiple stochastic

processes and the individual connectomes are all different.

• Although, in most animals, the brain connectivity varies between individuals, behaviour is often similar across species. Other words, despite differences in connectivity, most individuals under normal conditions are able to demonstrate similar functionalities.

Model of the nervous system

• Difference in connectivity - similarity on functionalitymeans that different connectomes include sufficient key structural features to produce a common repertoire of functionalities and behaviours.

• What are the key connectivity properties that define the network functionality?

• Motivated by this question, we developed a model of pair-wise connectivity in the nervous system of the hatchling Xenopus tadpole which, when combined with a spiking model of the Hodgkin-Huxley type, reliably reproduces appropriate motor behaviours mimicking the interaction with external environment.

• This biologically realistic model (VIRTUAL TADPOLE) can be used as a computational platform to crack a structure-function puzzle and find the key functional properties defining similarity of individual behaviours.

Xenopus tadpole spinal cord CPG

5mm longThere are 3 types of CPG neurons (ascending and descending interneurons (aIN and dIN) as well as commissural interneurons (cIN). Motor neurons (mn).There are 3 types of sensory pathway interneurons: touch skin sensors (RB), dorso-lateral ascending and commissural neurons (dla and dlc).

Spinal cord CPG

We start from studying the connectivityand spiking activity of spinal cordneuronal circuit in 2-day old Xenopustadpoles.~ 1500 neurons, 90K synapses and twobehaviours: swimming and struggling

2D plan of tadpole spinal cord

Can simple rules control development of a pioneer vertebrate neuronal network generating behavior?

A. Roberts, D. Conte, Mike Hull, R. Merrison-Hort, A. Azad, E.Buhl,R.Borisyuk, S.R. Soffe (2014)J of Neuroscience, 34: 608-621

Journal of NeuroscienceJournal of Neuroscience

From Connectome to Swimming Function

Experiment: swimming on touch

• Conductance based model of the Hodgkin-Huxley type.

• Connections between neurons are defined by the generated connectome.

• There are several characteristic electro-physiological features typical for tadpole swimming pattern (e.g. post-inhibitory rebound of dIN neurons, NMDA synapses).

• Model includes both electrical and synaptic connections, delays and noise in the parameters.

Roberts et al, J of Neurosci, 2014

Swimming Pattern

Stimulus is here

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Right

Sensory pathways

Li, Wagner, Porter, 2014 J Undergraduate Neuroscience Education

Touch skin

Touch head

Head pressure

Photo receptors

Initiation of swimming

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Touch skin population activity

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Left side motor neuron population activity and right side motor neuron population activity

STIMULUS

Anti-phasedIN

Anti-phasecIN

Bi-stability: Short-term stimulation moves system from a stable equilibrium to stable oscillations

time

Stopping

http://frogsaregreen.org/tag/froglet/

Can stop spontaneously and sink down to the ground

Swimming tadpoles stop when their head bumps into the water’s surface or objects like vegetation and the side of a dish

Roberts, Li, Soffe, 2010 Front Behavioral Neuroscince

https://www.youtube.com/watch?v=knlXTU1R_rE

Struggling (escaping) behaviour

https://www.youtube.com/watch?v=SJiwcRt-gQw

Roberts, Li, Soffe, 2010 Front Behavioral Neurosc

Struggling is a slower, stronger series of rhythmic alternating trunk flexions seen while tadpoles are grasped by predators.

Locomotor actions in the model of the nervous system

The repertoire of possible locomotor actions of the model includes:

• (a) start swimming (on sensory signal or spontaneously);

• (b) stop swimming (on sensory signal or spontaneously);

• (c) accelerating swimming;

• (d) struggling is not included yet

Roberts, Li, Soffe, 2010 Front Behavioral Neuroscience

Model of the nervous system

• We consider three sensory pathways: Skin Touch (ST), Head Touch (HT), and Head Pressure (HP). The hind brain decision making population processes the sensory information and sends a signal to CPG – to swim or not to swim.

• The total number of neuronal populations (neuronal types) K=12. The number of neurons in the model is about 2000. The total number of connections is about 100K.

• We design the biologically realistic model of connectivity and functionality. Building the model, we use a numerous data to reproduce activity patterns of initiation, continuation, acceleration and termination of swimming.

Tadpole Nervous SystemNervous system model includes sensory pathways, decision-making populations (hIN) and CPG neurons.

Connectome of the nervous system• A key part of our research is a pair-wise model of inter-neuronal

connectivity (connectome).

To find a connectome we combine two methods:

1) Developmental approach (Borisyuk et al., 2014) and

2) Probabilistic model (Ferrario et al., 2018).

1. For CPG and Skin touch sensory pathway neuronal populations there are available anatomical and morphological data which we use for the developmental approach (i.e. generate a set of biologically realistic axons and dendrite and find their intersections for synapse allocation).

2. For Head Touch and Head Pressure sensory pathway populations there is only a limited set of anatomical details. In these case we use a hypothetical approach based on similarities between neurons of different types to compose the probabilistic model (i.e. basing on available data we prescribe the probability of the pair-wise connection and using these probabilities we generate an adjacency matrix of connections).

3. For Decision Making population there are no anatomical data and we randomly prescribe the probability of connection.

Developmental ApproachComputational model generates a growing axon and synapses appear (with some probability) when the axon intersects a dendrite.

Distance from midbrain (µm)

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Neurons mapped onto 2D surface

Developmental Approach: Axon growth model

Longitudinal

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Dorso-Ventral gradient

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Coordinates

Growth angle

noise

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Where describe the chemical gradient cues which are universal for all axons while functions

describe the sensitivities of axon tip to each element of the gradient field.

Borisyuk et al., 2014 PLOS One

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Experiment

Model

Borisyuk et al., 2014 PLOS One

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Squared difference of experimentaland modelled projections

Tortuosity: Squared difference of experimental and modelledaverage tortuosity CC

Developmental approach: Model fitting and parametersFind the optimal values of five parameters: four sensitivities to gradients and the variance of random noise

Biologically realistic pattern of axons

Probabilistic model of the spinal cord

• Using the developmental approach we can generate multiple highly variable and nonhomogeneous connectomes of the spinal cord.

• Remarkably, ALL these connectomes, when projected to the functional spiking model, produce the swimming activity pattern.

• To simplify a process of the connectome generation (which includes a large and complex data set) we design a very simple meta-model expecting that this probabilistic model will reflect (generalise) structural properties of anatomical connectomes and show proper functioning (Ferrario et al., 2018).

Borisyuk et al., 2014 PLOS OneRoberts et al., 2014, J NeuroscDavis et al., 2018, Sci ReportsFerrario et al., 2018 eLife

Probabilistic Model of the spinal cord

To design the probabilistic model, we use a minimalistic approach. We assume that directed connections are represented by the matrix of independent Bernoulli random variables 𝑋𝑖𝑗, where Pr 𝑋𝑖𝑗 = 1is the probability of connection from 𝑖 to 𝑗.

We define the universal ordering of neurons in generated connectome to find the on-to-one correspondence between all anatomical connectomes.

To estimate the probability of connection we use averaging across 1000 anatomical

connectomes: Ƹ𝑝𝑖,𝑗 =𝑀𝑖𝑗

𝐾,

where 𝑀𝑖𝑗 is the number of connectomes with existing connection from 𝑖 to 𝑗 , 𝐾 =1000.

Visualization of the probability matrix

Ferrario et al., 2018 eLifie

ProbabilisticmodelWhite - no connectionRed - excitatory Blue - inhibitory

Colour intensity shows the probability of connection.

Distribution of in- and out-degrees

Mean values are the same for both models

Standard deviations are significantly larger for the anatomical model

Mean values and standard deviations are calculated from the probability matrix without simulations

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Connectome of the nervous system

Connections between ST sensory pathway and CPG neurons have been defined uing the anatomical model (developmental approach)

Connections between HT and HP sensory pathway neurons are based on similarities between neurons of different types and the probabilistic model

Connections between decision making neurons (xinor hIN) are randomly selected.

Matrix of connection probabilities. We use this matrix to generate a connectome

Decision-Making Population

Experiment: Time delay between skin touch and start of swimming varies in a wide range 20-150 ms.

Sustainable activity of the decision-making population builds up the ramping potential of CPG hdIN neurons.

When potential reaches the threshold, swimming starts.

Roberts, Borisyuk, et al., 2019, Proc Royal Soc B, Biol Sci Koutsikou et al., 2018, J Physiol

Trunk skin stimulation

Head skin stimulation

Sensory Stimuli and Motor Behaviour

Swim starting, stopping and starting

Swim acceleration and spontaneous stop

Conclusions

• We find that probabilistic connectomes that include some of the structure of anatomical connectomes reliably swim in all cases.

• Thus, we can derive an important conclusion that the two properties of the probabilistic model inherited from anatomical connectomes: position of neurons along the rostro-caudal coordinates and the frequency of connection appearance, are sufficient for swimming generation.

Conclusions• Probabilistic model allows analytical calculation of

some structural characteristics of the connectivity graph (i.e. the mean and standard deviation of in-and out-degrees, heterogeneity coefficients) directly from the probability matrix, without considerations of a particular (generated) connectome.

• We study how these structural characteristics relate to particular functional properties of the network. For instance, the average in- and out-degrees were used to predict the swimming period and to find the positions of reliably firing cINs.

Conclusion

• We demonstrate that in a number of cases, the model generates activity patterns similar to experiments (skin touch, head touch, head pressure).

• The model not only mimics experimental recordings but also can be used for prediction of some results and these predictions can be tested in real experiments.

• Remarkably(!), developing the model of the nervous system we do not take into account the system behaviour: we just generate a connectome and use it to produce the neuronal activities according to the model of spike generation.

• The behaviour emerges in the model as a response to the stimuli of different sensory modalities.

• To demonstrate this central result of our modelling we consider a scenario (a time sequence of stimuli) which modifies the model behaviour in a way which is similar to the tadpole behaviour in the natural environment.

Acknowledgement

Abul Azad

Alan Roberts Steve Soffe Debbie Conte Edgar Buhl

BRISTOL

Stella Koutsikou

St ANDREWSPLYMOUTH

Robert Merrison-Hort Wen-Chang Li

Andrea FerrarioMarius Varga