Intro

Model of ES (Electric stimulation) of spinal cord. It should enhance walking cycle with correct parameters. A proposed scheme is below.

Install

The project is compatible with python3 so when you start working on it you better have a python3 virtualenv. What is a virtualenv and how to work with it? Please read here. After you have got a virtualenv tool.

mkvirtualenv python3.5 -p /usr/bin/python3.5

After that you need to go the project's root and execute the below. Those are project's dependencies. For more information please read this.

pip install -r requirements.txt 

Also you need to have a custom NEST model hh_moto_5ht with your NEST installation.

Problem intro

During SCI(Spinal Cord Injury) a rat loses its ability to move hind limbs. It restores its ability to walk with ES(Epidural Stimulation) only after 5 weeks. This restoration of walking happens in sync with the appearance of LR(Late Response) in the EMG of muscles. MR(Middle Response) appears during the 2nd week. Injection of 5-HT enhances the same effect of walking restoration almost immediately when applied directly to the spinal cord. This allows us to derive that during first two weeks the monosynaptic circuit or the first level restores and it brings MR to the EMG. During five weeks the polysynaptic circuit restores and brings LR to the EMG.

Some biology details

Muscle is innervated by motoneurons (efferents), sensory neurons (afferents). They can be classified.

MotoNeurons

• Alpha motoneurons. Innervate muscles. Convey signals to contraction/stretch.
• Gamma motoneurons. Innervate muscle spindles. Convey signals to contraction/stretch.

Sensory neurons

• Group 1.
  • Group 1a. Innervates muscle spindles. Conveys input about contraction/stretch. Fast. It excites its agonist, inhibits antagonist.
  • Group 1b. Innervates junction between muscle and its tendon. Not presented on images. Conveys input about its own contraction/stretch. It inhibits its agonist.
• Group 2. Innervates muscle spindles. Conveys input about contraction/stretch. Slow. It excites its agonist, inhibits antagonist. When a muscle is relaxed, all of its afferents are zero. Lets contract the muscle. Information about its contraction/stretch is conveyed with 0.2s delay here comparing to Group 1a.

About computational model of Moraud and Marco 2016

They research the influence of Group 1a, Group 2 afferents only. Their simple model is presented at the figure 1A. Afferents are represented as Frequency Generators. They generate different activity for different environments. Those activities are predefined and were gotten by recording from real rat's afferents in those different environments. Model's validation:

• Serotonin-mediated modulation of motoneurons is conveyed by reducing the conductance of potassium-calcium gated ion channels. They report that the lack of 5HT modulation can be compensated by increasing ES frequency. Figure 8A.
• Variation of single ES intensity as on the Figure 1E.
• During locomotion. Increase of ES frequency led to linear increase in the mean firing rate of Ia and II afferents comparable to the increase of single ES. Temporal profiles of afferent firing rates were preserved. The linear increase in afferents led to a linear increase of motoneurons during their active phase. Increase of ES amplitude led to direct recruitment of motoneurons and disruption of alternation between flexor and extensor. See Figure 2A, 2B. How did the increase of ES frequency lead to the increase of afferents frequency, if afferents were represented as Frequency generators?
• During locomotion. Protocols of ES: 40 hz + 1.2 motor threshold amplitude, 80 hz + 1.2 motor threshold amplitude, 40 hz + 1.4 motor threshold amplitude, - must behave as Figure 2C.
• During locomotion. The duration of gait cycle must be sync with the step speed as on Figure 2D.
• Each phase of gait is independent of the other e.g. we can modulate ES each phase independently in order to balance step heights for example. The collateral of this is a full balance of steps. So, independent ES frequencies are much better.

General questions:

• Temporal profile in firing rate?
• Frequency harmonics?

Network parameters

Neurons

Numbers below are per muscle e.g. for example the flexor. For the antagonist muscle numbers are the same. Initial numbers were taken from Moraud 2016.

MotoNeurons

• Number: 169
• Type: HH model with: sodium, potassium, calcium, and potassium-calcium. For the start we would use only: sodium, potassium.

1A fibers

• Number: 60
• Type: IntFire1 with tau=0.5, refrac=1
• Firing rate: Ia firing rate = 50 + 2*stretch + 4.3*sign(strVelocity)*|strVelocity|^0.6 + 50*EMG

1B fibers

• ?

2 fibers

• Number 60
• Type: - Type: IntFire1 with tau=0.5, refrac=1
• Firing rate: II firing rate = 80 + 13.5*stretch + 20*EMG

Cutaneous fibers

• ?

1A Inhibitory InterNeurons

• Number: 196
• Neuron Type: IntFire4 with taue=0.5, taui1=5, taui2=10, taum=30
• Nest Type: iaf_psc_alpha * with tau_syn_ex=0.5, tau_syn_in=5, tau_m=30

2 Excitatory InterNeurons

• Number: 196
• Neuron Type: IntFire4 with taue=0.5, taui1=5, taui2=10, taum=30
• Nest Type: iaf_psc_alpha * with tau_syn_ex=0.5, tau_syn_in=5, tau_m=30

Connections

flexor 1A-fibers - flexor MotoNeurons

Each motoneuron connected to all 1A fibers without any tricks and randomness.

• from SynFlexFlex.hoc:53-66, M_Cell.hoc:105-125
• nest connection type: all-to-all.
• Neuron type: ExpSyn with e=0, tau=0.5.
• Nest type: none specific. Please see below.
• delay: 2 + Normal(0, 0.3 || 0.03) ms. In the paper 0.3 but in src 0.03. Ia+taur.normal(0,0.03).
• weight: 0.052608, hi_motor_S = 0.0411 + 0.0411 * 0.28.

flexor 1A-fibers - flexor 1A-Inhibitory InterNeurons

Connect each interneuron 62 times with all fibers randomly picking the fiber each connection.

• from SynFlexFlex.hoc:68-78
• nest connection type: fixed-indegree 62 FromAll_IAf_ToOne_IAint. InterNeurons are 'in'. Uniform.
• delay: 2 + Normal(0, 0.03) ms. Ia+taur.normal(0,0.03)
• weight: 0.0175, hi= 0.0175

flexor 2-Excitatory InterNeurons - flexor MotoNeurons

• from SynFlexFlex.hoc:81-92, M_Cell.hoc:128-140
• nest connection type: fixed-indegree 116 FromAll_EXIN_ToOne_MN. MotoNeurons are 'in'. Uniform.
• Neuron type: ExpSyn with e=0, tau=0.5.
• Nest type: none specific. Please see below.
• delay: 1ms, tausyn=1
• weight: 0.00907, hi_motor_Ex= (hi_motor_S*FromAll_IAf_ToOne_MN/FromAll_EXIN_ToOne_MN)/3

flexor 2-Fibers - flexor 2-Excitatory InterNeurons

• from SynFlexFlex.hoc:95-106
• nest connection type: fixed-indegree 62 FromAll_IIf_ToOne_EXIN. InterNeurons are 'in'. Uniform.
• delay: 3 + Normal(0, 0.03) ms, II+taur.normal(0,0.03)
• weight: 0.0175, hi= 0.0175

flexor 2-Fibers - flexor 1A-Inhibitory InterNeurons

• from SynFlexFlex.hoc:108-119
• nest connection type: fixed-indegree 62 FromAll_IIf_ToOne_IAint. InterNeurons are 'in'. Uniform.
• delay: 3 + Normal(0, 0.03) ms, II+taur.normal(0,0.03)
• weight: 0.0175, hi= 0.0175

extensor 1A-Inhibitory InterNeurons - flexor 1A-Inhibitory InterNeurons

• from SynExtFlex.hoc:6-18
• nest connection type: fixed-indegree 100 FromAll_IAint_ToOne_IAint. Flexor InterNeurons are 'in'. Uniform.
• delay: 1ms, tausyn=1
• weight: -0.007, -(lo=0.007)

flexor 1A-Inhibitory InterNeurons - extensor 1A-Inhibitory InterNeurons

• from SynExtFlex.hoc:20-30
• nest connection type: fixed-indegree 100 FromAll_IAint_ToOne_IAint. Extensor InterNeurons are 'in'. Uniform.
• delay: 1ms, tausyn=1. Warning, in the sources this delay is missing
• weight: -0.007, -(lo=0.007)

extensor 1A-Inhibitory InterNeurons - flexor MotoNeurons

Every motoneuron connected 232 times randomly to the pool of interneurons. There is a very small possibility that it will connect 232 times to the same interneuron.

• from SynExtFlex.hoc:32-44, M_Cell.hoc:141-147
• nest connection type: fixed-outdegree 232 FromAll_IAint_ToOne_MN. MotoNeurons are 'out'. Uniform.
• Neuron type: Exp2Syn with e=-75, tau1=1.5, tau2=2.
• Nest type: none specific. Please see below.
• delay: 1ms, tausyn=1
• weight: 0.0023, lom=0.0023

flexor 1A-Inhibitory InterNeurons - extensor MotoNeurons

• from SynExtFlex.hoc:46-57, M_Cell.hoc:141-147
• nest connection type: fixed-indegree 232 FromAll_IAint_ToOne_MN. MotoNeurons are 'in'. Uniform.
• type: Exp2Syn with e=-75, tau1=1.5, tau2=2.
• Nest type: none specific. Please see below.
• delay: 1ms, tausyn=1
• weight: 0.0023, lom=0.0023

Neuron to Nest

Problems:

• Nest models do not have dual exponential as time constants. Only aplha functions.
• Neuron neurons mostly are normalized. It means that V_m value range is [0, 1]mV instead of usual [-70, 40]mV.
• Nest synapses implement inhibition by negative weights. Neuron synapses implement inhibition with Reversal Potential and their weights are positive. So it's a separate task to transform inhibitory positive weights from Neuron to negative weights of Nest.

IntFire4 does not have a counterpart in Nest. We use Nest's iaf_psc_alpha which uses an alpha function instead of a dual exponential for inhibitory tau. That is why taui2=10 isn't used. ExpSyn does not have a counterpart in Nest. Example how can we achieve it for excitation only. Lets take an HH neuron with an ExpSyn on it e=0, tau=0.8. In Nest it would be:

neurons = nest.Create("hh_psc_alpha")
nest.CopyModel("static_synapse","excitatory",{"weight":1.,  "delay":1.5})
nest.Connect(neurons, ispikes, syn_spec="excitatory")

According to this article.

Exp2Syn The same as above. Lets take an HH neuron with an Exp2Syn on it e=-75.0, tau1=0.99, tau2=1.0. In Nest it would be (for excitation only):

neurons = nest.Create("hh_psc_alpha", {'tau_syn_in':1.})
nest.CopyModel("static_synapse","inhibitory",{"weight":200., "delay":3.0})
nest.Connect(neurons, ispikes, syn_spec="inhibitory")