# Thalamo-cortical connections

In this post I will show how to build arbitrary custom connections in PyNN. We will illustrate the general technique in the particular case of the Troyer model. In the Troyer model the connections from the LGN to the cortex are determined with a gabor-profile therefore I am going to describe the required functions to achieve such an aim.

In the PyNN documentation we find that one of the ways of implementing arbitrary connectivity patterns is to use the FromListConnector utility. In this format we have to construct a list of tuples with a tuple for each connection. In each tuple we need to include the index of the source neuron (the neuron from which the synapse originates), the index of the target neuron (the neuron into which the synapse terminates), the weight and the delay. For example (0, 1, 5, 0.1) would indicate that we have a connection from the neuron 0 to the neuron 1 with a synaptic weight of 5 and a delay of 0.1.

In the light of the explanation above we need to construct a function that is able to construct a list with the appropriate weights given a target and a source populations. In order to start moving towards this goal we will first write a function that connects a given neuron in the target population to all the neurons in the source population. We first present the function here bellow and we will explain it later:

def lgn_to_cortical_connection(cortical_neuron_index, connections, lgn_neurons, n_pick, g, delay, polarity, sigma, gamma, phi, w, theta, x_cortical, y_cortical): """ Creates connections from the LGN to the cortex with a Gabor profile. This function adds all the connections from the LGN to the cortical cell with index = cortical_neuron_index. It requires as parameters the cortical_neruon_index, the current list of connections, the lgn population and also the parameters of the Gabor function. Parameters ---- cortical_neuron_index : the neuron in the cortex -target- that we are going to connect to connections: the list with the connections to which we will append the new connnections lgn_neurons: the source population n_pick: How many times we will sample per neuron g: how strong is the connection per neuron delay: the time it takes for the action potential to arrive to the target neuron from the source neuron polarity: Whether we are connection from on cells or off cells sigma: Controls the decay of the exponential term gamma: x:y proportionality factor, elongates the pattern phi: Phase of the overall pattern w: Frequency of the pattern theta: Rotates the whole pattern by the angle theta x_cortical, y_cortical : The spatial coordinate of the cortical neuron """ for lgn_neuron in lgn_neurons: # Extract position x, y = lgn_neuron.position[0:2] # Calculate the gabbor probability probability = polarity * gabor_probability(x, y, sigma, gamma, phi, w, theta, x_cortical, y_cortical) probability = np.sum(np.random.rand(n_pick) < probability) # Samples synaptic_weight = (g / n_pick) * probability lgn_neuron_index = lgn_neurons.id_to_index(lgn_neuron) # The format of the connector list should be pre_neuron, post_neuron, w, tau_delay if synaptic_weight > 0: connections.append((lgn_neuron_index, cortical_neuron_index, synaptic_weight, delay))

The first thing to note from the function above are its arguments. It contains the source population and the particular target neuron that we want to connect to. It also contains all the connectivity and gabor-function related parameters. In the body of the function we have one loop over the whole source population that decides whether we add a connection from a particular cell or not. In order to decide if we add a connection we have to determine the probability from the gabor function. Once we have this we sample n_pick times and add a weighted synaptic weight accordingly to the list for each neuron.

In the function above we have the values of the gabor function passed as arguments. However, the values of each gabor function depend on the nature of the cell of the target population. In the light of this we will construct another function that loops over the target population and extracts the appropriate gabor values for each function in this population. We again present the function and then explain it:

def create_lgn_to_cortical(lgn_population, cortical_population, polarity, n_pick, g, delay, sigma, gamma, phases, w, orientations): """ Creates the connection from the lgn population to the cortical population with a gabor profile. It also extracts the corresponding gabor parameters that are needed in order to determine the connectivity. """ print 'Creating connection from ' + lgn_population.label + ' to ' + cortical_population.label # Initialize connections connections = [] for cortical_neuron in cortical_population: # Set the parameters x_cortical, y_cortical = cortical_neuron.position[0:2] cortical_neuron_index = cortical_population.id_to_index(cortical_neuron) theta = orientations[cortical_neuron_index] phi = phases[cortical_neuron_index] # Create the connections from lgn to cortical_neuron #lgn_to_cortical_connection(cortical_neuron_index, connections, lgn_population, n_pick, g, polarity, sigma, #gamma, phi, w, theta, x_cortical, y_cortical) lgn_to_cortical_connection(cortical_neuron_index, connections, lgn_population, n_pick, g, delay, polarity, sigma, gamma, phi, w, theta, 0, 0) return connections

This function requires as arguments the source and target populations as well as the necessary parameters that characterize each cell connectivity: orientation and phase. In the body of the function we have a loop over the cortical population that extracts the relevant parameters -position, orientation and phase- and then calls the function that we already describe previously in order to create the connectivity from the source population to the cell in place.

So now we have the necessary functions to construct a list. Now, we can use FromListConnector to transform the list into a connector. And the use this to define a Projection. We define both the excitatory and inhibitory connections. We abstract this complete set into the following function:

def create_thalamocortical_connection(source, target, polarity, n_pick, g, delay, sigma, gamma, w, phases, orientations, simulator): """ Creates a connection from a layer in the thalamus to a layer in the cortex through the mechanism of Gabor sampling """ # Produce a list with the connections connections_list = create_lgn_to_cortical(source, target, polarity, n_pick, g, delay, sigma, gamma, phases, w, orientations) # Transform it into a connector connector = simulator.FromListConnector(connections_list, column_names=["weight", "delay"]) # Create the excitatory and inhibitory projections simulator.Projection(source, target, connector, receptor_type='excitatory') simulator.Projection(source, target, connector, receptor_type='inhibitory')

With this we can create in general connections from one target population to the other. We can even change change the gabor function for whatever we want if we want to experiment with other connectivity patterns. Finally we present down here an example of a sampling from a Gabor function with the aglorithm we just constructed:

So in the image we show in the left the sampling from the ideal Gabor function in the right.

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