Vertical System (VS) tangential cells network model (Trousdale et al. 2014)

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Accession:155727
Network model of the VS tangential cell system, with 10 cells per hemisphere. Each cell is a two compartment model with one compartment for dendrites and one for the axon. The cells are coupled through axonal gap junctions. The code allows to simulate responses of the VS network to a variety of visual stimuli to investigate coding as a function of gap junction strength.
Reference:
1 . Trousdale J, Carroll SR, Gabbiani F, Kresimir Josic K (2014) Near-Optimal Decoding of Transient Stimuli from Coupled Neuronal Subpopulations J Neurosci 34(36):12206-12222
Model Information (Click on a link to find other models with that property)
Model Type: Realistic Network; Axon; Synapse; Dendrite;
Brain Region(s)/Organism:
Cell Type(s): Fly vertical system tangential cell;
Channel(s):
Gap Junctions: Gap junctions;
Receptor(s): Nicotinic; GabaA;
Gene(s):
Transmitter(s): Acetylcholine; Gaba;
Simulation Environment: Python;
Model Concept(s): Activity Patterns; Spatio-temporal Activity Patterns; Simplified Models; Invertebrate; Connectivity matrix;
Implementer(s): Gabbiani, F; Trousdale, James [jamest212 at gmail.com];
Search NeuronDB for information about:  Nicotinic; GabaA; Acetylcholine; Gaba;
'''
Function:     sim_vs_net

Arguments:    param_dict - Parameter dictionary, expected variables and definitions below. See run_time_vars at end of script for 
                           default parameter values 
              movie (optional) - The movie to supply as input to the system. If a movie is not provided, one will be
                                 generated using the subdictionary param_dict['image']
                                 
              Expected contents of the dictionary param_dict:
              
              g_gap - Axo-axonal gap junction coupling strength. (nS)
              g_inh_scale - Scaling factor for the negative conductance electrical coupling between VS1 and VS10. The
                            strength of the connection is set to g_gap*g_inh_scale. (no units)
              g_L_dend - Dendritic leak conductance. (nS)
              g_L_axon - Axonal leak conductance. (nS)
              g_axon_dend - Axo-dendritic electric coupling strength. (nS)
              g_exc_in - Excitatory input conductance strength. (nS)
              g_inh_in - Inhibitory input conductance strength. (nS)
              E_E - Excitatory input reversal potential. (mV)
              E_I - Inhibitory input reversal potential. (mV)
              tau_m - Axonal and dendritic membrane time constant. (ms)
              noise_std - Intensity of intrinsic white noise fluctuations in axonal and dendritic compartments. (pA) 
              tau_hp - Time constant of the high-pass filter component of the Reichardt detectors. (ms)
              tau_lp - Time constant of the low-pass filter component of the Reichardt detectors. (ms)
              rot_angle - Horizontal rotation axis. Rotation about this axis induces apparent clockwise rotation of the
                          external world to the fly (i.e., it corresponds to a counter-clockwise roll of the fly about
                          this axis). (deg)
                          For the stripe stimuli, rot_angle is the azimuthal center of the stripe stimulus. (deg)
              deg_per_ms - Speed of the clockwise rotation about the horizontal axis at the angle rot_angle. (deg/ms)
                           For the stripe stimulus, this is the speed of downward motion. (deg/ms)
              T - Total simulation duration. (ms)
              dt_sim - Time step of integration for the membrane potentials. (ms)
              dt_im - Time step used for generating the movie. The induced conductances are then linearly interpolated
                      to the time step dt_sim for integration.
              image - (required if no movie is supplied) Image parameter dictionary, with parameters
                          type - Image type, determines which image generation module is called
                          x_pixels - Number of pixels along the horizontal direction
                          y_pixels - Number of pixels along the vertical direction
              
           
Output:       V_axon_L - Time course of the left-side VS axonal responses at times in the interval [0,T-dt_sim] with 
                         time step dt_sim. Dimension is (10,T/dt_sim), and V_axon_L[i,:] is the response of the neuron
                         VS(i+1). (mV)
              V_axon_R - Same as V_axon_L, but for right-side VS axonal responses.
           
Description:  Simulate the blow fly VS network. The method of simulation is adapted from that of Borst and Weber (2011)
              "Neural action fields for optic flow based navigation: a simulation study of the fly lobula plate 
              network...". Includes only the portion corresponding to the left- and right-side VS network, and the
              "repulsive" coupling which approximates impact of the Vi, Vi2 cells. Generates a movie corresponding to
              a rotation of a generated spherical image about the given horizontal axis. Alternatively, can take as an
              argument a user-generated movie. For details, refer to:
                    Near optimal decoding of transient stimuli from coupled neuronal subpopulations
                    by James Trousdale, Sam Carroll, Fabrizio Gabbiani, and Kresimir Josic.
                    J. Neurosci. ??:??-??..

Authors:     James Trousdale - jamest212@gmail.com
             * Adapted from MATLAB code provided by Drs. Y. Elyada and A. Borst
'''


__all__ = ["sim_vs_net","reichardt_loc","reichardt_filter"]

import numpy as np
from scipy.interpolate import interp1d as interp1d
from scipy.signal import sepfir2d as sepfir2d
from utilities import ang_dist,repmat_3d,my_2d_hist

#                                             #
#                                             #
###### definition of function sim_vs_net ######
#                                             #
#                                             #

def sim_vs_net(param_dict,movie=[]):
    
    rs_ctr = np.array([10, 26, 42, 58, 74, 90, 106, 122, 138, 154]) # Gaussian receptive field horizontal centers
    sig_th = 12; # Gaussian receptive field horizontal standard deviation (degrees)
    sig_phi = 60; # Gaussian receptive field vertical standard deviation (degrees)
    
    
    T = param_dict['T']
    dt_sim = param_dict['dt_sim']
    dt_im = param_dict['dt_im']
    
    g_gap = float(param_dict['g_gap'])
    g_inh = g_gap*param_dict['g_inh_scale']
    g_L_dend = float(param_dict['g_L_dend'])
    g_L_axon = float(param_dict['g_L_axon'])
    g_axon_dend = float(param_dict['g_axon_dend'])
    
    g_exc_in = float(param_dict['g_exc_in'])
    g_inh_in = float(param_dict['g_inh_in'])
    E_E = float(param_dict['E_E'])
    E_I = float(param_dict['E_I'])
    
    tau_m = float(param_dict['tau_m'])
    noise_std = float(param_dict['noise_std'])
    
    tau_hp = float(param_dict['tau_hp'])
    tau_lp = float(param_dict['tau_lp'])
    
    
    # Check to see if a movie was provided. If one was, use it. If not, generate one
    # according to the dictionary param_dict['image']. At minimum, param_dict['image']
    # must contain a variable 'type' indicating the image type (i.e., white, bar or natural),
    # and the horizontal/vertical dimensions.
    # Refer to the respective image generation modules for more details.
    if np.size(movie) == 0:
        import rotate_image as ri
        
        # Check the image type, call the appropriate image generation module, then generate the movie
        if param_dict['image']['type'] == 'white':
            image = ri.white_image(param_dict['image']['x_pixels'],param_dict['image']['y_pixels'],
                                   param_dict['image']['upsample'])
            movie = ri.rotate_sphere(image,param_dict['rot_angle'],param_dict['deg_per_ms'],T,dt_im)
        elif param_dict['image']['type'] == 'bar':
            image = ri.bar_image(param_dict['image']['x_pixels'],param_dict['image']['y_pixels'],
                                 param_dict['image'])
            movie = ri.rotate_sphere(image,param_dict['rot_angle'],param_dict['deg_per_ms'],T,dt_im)
        elif param_dict['image']['type'] == 'natural':
            image = ri.natural_image(param_dict['image']['x_pixels'],param_dict['image']['y_pixels'],
                                     param_dict['image']['image_lib_path'])
            movie = ri.rotate_sphere(image,param_dict['rot_angle'],param_dict['deg_per_ms'],T,dt_im)
        elif param_dict['image']['type'] == 'stripe':
            movie = ri.vert_strip(param_dict['image']['x_pixels'],param_dict['image']['y_pixels'],
                                  param_dict['rot_angle'],param_dict['image']['strip_rad'],
                                  param_dict['image']['freq_s'],param_dict['deg_per_ms'],T,dt_im)
            
    # Logarithmic scaling of the movie accounts for the signal transduction which occurs at the level of 
    # retinal photoreceptors
    movie = np.log10(movie+1) - np.mean(np.log10(movie+1))
    
    
    # Define the vectors of membrane conductances and capacitances. 
    # Indexing of cells is consistent throughout, and is as follows
    #     Even indices --- dendrites, in increasing order (so 0 corresponds to VS1 dendrite, 2 to VS2 dendrite, etc.)
    #     Odd indices --- axons, in increasing order (so 1 corresponds to VS1 axon, etc.)
    g_L = np.zeros(20) 
    g_L[np.arange(0,20,2)] = g_L_dend
    g_L[np.arange(1,20,2)] = g_L_axon
    C_m = tau_m*g_L 
    
    
    # Construct the electrical coupling weight matrix. Dimension is (20,20), corresponding to the total number of
    # axons and dendrites on one side (the two sides are not coupled presently). This matrix allows the system to be
    # written in the form
    #            V' = GV + noise/input
    G = np.zeros((20,20))
    
    # Couple axons and dendrites of each cell
    for i in np.arange(1,20,2): 
        G[i-1,i] = g_axon_dend
        G[i,i-1] = g_axon_dend
        
    # Couple consecutive axons
    for i in np.arange(3,20,2):
        G[i-2,i] = g_gap;
        G[i,i-2] = g_gap;
    
    # Couple the VS1 and VS10 dendrites via a "repulsive" synapse (electrical synapse with negative conductance)
    # as in Weber, et al. (2008) "Eigenanalysis of a neural network..." --- emulates the effect of including the
    # inhibitory spiking neurons Vi, Vi2.
    G[0,18] = -g_inh; G[18,0] = -g_inh;
    
    # Normalize the diagonal of the weight matrix by the negatives of couplings with other compartments, reflecting the
    # bidirectionality of the gap junctions.
    for i in range(20):
        G[i,i] = -np.sum(G[i,:])-g_L[i]
        
    
    # Call the reichardt_loc function to get the pixel coordinates of the Reichardt detectors
    reichardt_locs = reichardt_loc(np.size(movie,1),np.size(movie,0))
    
    # Apply the Reichardt detector filtering to the movie
    (reich_up,reich_down) = reichardt_filter(movie,reichardt_locs,tau_hp,tau_lp,dt_im)
    
    # Smooth the upward and downward components of the Reichardt detector-filtered image by a 1x3 box filter, 
    # separately in each dimension
    conv_kernel_1d = np.ones(3)/3
    for i in range(np.size(reich_up,2)):
        reich_up[:,:,i] = sepfir2d(reich_up[:,:,i],conv_kernel_1d,conv_kernel_1d)
        reich_down[:,:,i] = sepfir2d(reich_down[:,:,i],conv_kernel_1d,conv_kernel_1d)
        
        
        
    # Compute receptive fields as a product of Gaussians corresponding to horizontal and vertical components.
    recf_L = np.zeros((10,np.size(movie,0),np.size(movie,1)))
    recf_R = np.copy(recf_L)
    
    # phi_exp contains the Gaussian for the vertical component of the receptive fields.
    phi_exp = np.reshape(np.exp(-(np.linspace(-90,90,np.size(movie,0),endpoint=False)**2)/(2*sig_phi**2)),
                         (np.size(movie,0),1))
    
    theta_vals = np.linspace(0,360,np.size(movie,1),endpoint=False) # Discretization of horizontal angles
    
    # For each cell, we compute the Gaussian corresponding to the horizontal component of the receptive field, with
    # the usual difference in the exponent replaced with an angular distance, and mean equal to the horizontal center
    # of each cells receptive field. The horizontal component is reshaped to be dimension (1,x_pixels), and is
    # multiplied with the (y_pixels,1) vertical component, giving the dimension (y_pixels,x_pixels) receptive field,
    # matching the dimension of the movie being supplied as input. The receptive field is then normalized to sum to 1
    # over the pixels which correspond to the location of a Reichardt detector --- pixels without a detector are not
    # registered, and have no impact on the activity of the system.
    for n in range(10):
        th_exp_L = np.reshape(np.exp(-ang_dist(theta_vals,-rs_ctr[n])**2/(2*sig_th**2)),(1,np.size(movie,1)))
        th_exp_R = np.reshape(np.exp(-ang_dist(theta_vals,rs_ctr[n])**2/(2*sig_th**2)),(1,np.size(movie,1)))
        
        recf_L[n,:,:] = np.dot(phi_exp,th_exp_L) 
        recf_L[n,:,:] = recf_L[n,:,:]/np.sum(recf_L[n,:,:]*reichardt_locs)
        
        recf_R[n,:,:] = np.dot(phi_exp,th_exp_R)
        recf_R[n,:,:] = recf_R[n,:,:]/np.sum(recf_R[n,:,:]*reichardt_locs)
        
    
    # Before beginning the simulation, compute the number of time steps at the coarse scale (time step for the
    # image rotation), and the fine scale (time step for the integration of the membrane potentials).
    coarse_steps = int(T/dt_im)+1
    time_steps = int((T-dt_sim*0)/dt_sim)
    
    t_range = np.linspace(0,T,time_steps,endpoint=False)
    
    g_exc_L = np.zeros((20,time_steps))
    g_inh_L = np.zeros((20,time_steps))
    g_exc_R = np.zeros((20,time_steps))
    g_inh_R = np.zeros((20,time_steps))
    
    # Compute the synaptic conductances induced by the image generation for each cell
    for n in range(10):
        # For each side, first duplicate the receptive field weight matrix computed above along the time dimension,
        # then take a matrix dot product (i.e., a Hadamard product) of the duplicated receptive field with the 3-D
        # array of Reichardt detector outputs, taken at each time step. Summing these values at each time step give
        # the excitatory (downward motion) and inhibitory (upward motion) input conductances.
        rep_recf = repmat_3d(recf_L[n,:,:],coarse_steps)
        g_exc_L_coarse = np.sum(np.sum(rep_recf*reich_down,axis=0),axis=0)
        g_inh_L_coarse = np.sum(np.sum(rep_recf*reich_up,axis=0),axis=0)
    
        rep_recf = repmat_3d(recf_R[n,:,:],coarse_steps)
        g_exc_R_coarse = np.sum(np.sum(rep_recf*reich_down,axis=0),axis=0)
        g_inh_R_coarse = np.sum(np.sum(rep_recf*reich_up,axis=0),axis=0)
        
        # Conductances were initially computed at the coarse timescale. First use interp1d to define the linear
        # interpolating function for the excitatory and inhibitory conductances on each side...
        interp_g_exc_L = interp1d(np.linspace(0,T,coarse_steps,endpoint=True),g_exc_L_coarse)
        interp_g_inh_L = interp1d(np.linspace(0,T,coarse_steps,endpoint=True),g_inh_L_coarse)
        
        interp_g_exc_R = interp1d(np.linspace(0,T,coarse_steps,endpoint=True),g_exc_R_coarse)
        interp_g_inh_R = interp1d(np.linspace(0,T,coarse_steps,endpoint=True),g_inh_R_coarse)
        
        # ... then compute the conductances at each of the finer scale time steps using the linear interpolation.
        g_exc_L[2*n,:] = interp_g_exc_L(t_range)*g_exc_in
        g_inh_L[2*n,:] = interp_g_inh_L(t_range)*g_inh_in
    
        g_exc_R[2*n,:] = interp_g_exc_R(t_range)*g_exc_in
        g_inh_R[2*n,:] = interp_g_inh_R(t_range)*g_inh_in
    
    
    # Pre-generate the white noise injected in to each dendritic and axonal compartment.    
    dW_L = 1.0/np.sqrt(dt_sim)*np.random.randn(20,time_steps)*noise_std
    dW_R = 1.0/np.sqrt(dt_sim)*np.random.randn(20,time_steps)*noise_std
    
    
    V_L = np.zeros(np.shape(g_exc_L))
    V_R = np.zeros(np.shape(g_exc_R))
    
    # The incoming synaptic current is the sum of the synaptic conductances, weighted by the reversal potentials.
    syn_curr_L = g_exc_L*E_E + g_inh_L*E_I
    syn_curr_R = g_exc_R*E_E + g_inh_R*E_I
    
    
    # Integrate the membrane potentials via an implicit, first order scheme
    H = np.diag(C_m/dt_sim) - G
    for i in range(1,np.size(syn_curr_L,1)):
        V_L[:,i] = np.linalg.solve(H + np.diag(g_exc_L[:,i]) + np.diag(g_inh_L[:,i]),
                                   syn_curr_L[:,i] + V_L[:,i-1]*C_m/dt_sim + np.sqrt(tau_m)*dW_L[:,i])
        V_R[:,i] = np.linalg.solve(H + np.diag(g_exc_R[:,i]) + np.diag(g_inh_R[:,i]),
                                   syn_curr_R[:,i] + V_R[:,i-1]*C_m/dt_sim + np.sqrt(tau_m)*dW_R[:,i])
        
        
    # Return only the axonal potentials (could easily be changed to return dendritic potentials as well...)
    V_axon_L = V_L[np.arange(1,20,2),:]
    V_axon_R = V_R[np.arange(1,20,2),:]
    return (V_axon_L,V_axon_R)




'''
Function:     reichardt_loc

Arguments:    x_pixels - number of horizontal pixels which will be utilized in images and movies
              y_pixels - number of vertical pixels which will be utilized in images and movies
           
Output:       A dimension (y_pixels,x_pixels) binary array, with 1's corresponding to pixels containing a Reichardt
              detector.
           
Description:  Given horizontal/vertical image dimensions x_pixels/y_pixels, spreads 10,000 Reichardt detectors 
              approximately evenly on the surface of a sphere using the Golden Section spiral algorithm. For reference, 
              see Patrick Boucher's blog post at http://www.softimageblog.com/archives/115 (posted October 4, 2006; 
              last accessed June 25, 2014). For 10,000 detectors, the approximate separation between detectors is 
              between 2 and 3 degrees, in the biologically realistic range.

Authors:     James Trousdale - jamest212@gmail.com
             * Adapted from code provided by Patrick Boucher
'''


#                                                #
#                                                #
###### definition of function reichardt_loc ######
#                                                #
#                                                #

def reichardt_loc(x_pixels,y_pixels):
    
    num_detectors = 10000
    
    theta_edges = np.linspace(0,360,x_pixels+1,endpoint=True)
    phi_edges = np.linspace(0,180,y_pixels+1,endpoint=True)
    
    # Compute the rectangular coordinates (x,y,z) of the location on the unit sphere of the Reichardt detectors.
    inc = np.pi*(3-np.sqrt(5))
    off = 2.0/num_detectors
    i_vec = np.array(range(num_detectors))
    y = i_vec*off - 1 + off/2
    r = np.sqrt(1-y*y)
    phi = i_vec*inc
    x = np.cos(phi)*r
    z = np.sin(phi)*r
    
    # Compute spherical coordinates of the pixels which correspond to the location of a Reichardt detector.
    theta_coords = np.mod(np.arctan2(y,x),2*np.pi)*180/np.pi
    phi_coords = np.arccos(z)*180/np.pi
    
    # Return an array which has a 1 in the entries corresponding to pixels with a detector
    return my_2d_hist(theta_coords,phi_coords,theta_edges,phi_edges)
        
        


'''
Function:     reichardt_filter

Arguments:    movie - a movie (here, generated by rotation of an image about an axis) of dimension
                      (y_pixels,x_pixels,time_steps)
              reichard_locs - binary array of dimension (y_pixels,x_pixels), with 1 corresponding to a pixel containing
                              a Reichardt detector
              tau_hp - time constant of the high-pass filter in each half of the Reichardt detector
              tau_lp - time constant of the low-pass filter in each half of the Reichardt detector
              dt - time step of integration for the application of the Reichardt filtering
           
Output:       reich_up - Upward component of motion detected by the Reichardt filtering
              reich_down - Downward component of motion detected by the Reichardt filtering
           
Description: The reichardt_filter function filters the given movie through an array of Reichardt detectors, returning
             components corresponding to upward and downward motion. The type of detector implemented here makes use of
             a total of four filters --- two high pass and two low pass. For details, see Borst and Weber (2011) "Neural Action
             Fields for Optic Flow Based Navigation: A Simulation Study of the Fly Lobula Plate Network", 
             PLoS ONE 6(1): e16303. doi:10.1371/journal.pone.0016303.

Authors:     James Trousdale - jamest212@gmail.com
             * Adapted from code provided by Drs. Y. Elyada and A. Borst
'''

#                                                   #
#                                                   #
###### definition of function reichardt_filter ######
#                                                   #
#                                                   #


def reichardt_filter(movie,reichardt_locs,tau_hp,tau_lp,dt):
    
    y_res = 180.0/np.size(movie,0) # Vertical size of pixels (in degrees).
    
    dims = np.array(np.shape(movie))
    
    high_pass = np.zeros(dims)
    low_pass = np.zeros(dims)
    reich = np.zeros(dims)
    
    high_pass_const = dt/(tau_hp + dt)
    low_pass_const = dt/(tau_lp + dt)
    
    # Perform high- and low-pass filtering in time of the movie
    for i in range(1,np.size(movie,2)):
        high_pass[:,:,i] = high_pass_const*movie[:,:,i] + (1-high_pass_const)*high_pass[:,:,i-1]
        low_pass[:,:,i] = low_pass_const*movie[:,:,i] + (1-low_pass_const)*low_pass[:,:,i-1]
    high_pass = movie - high_pass
    
    
    # Each detector is composed of two detector subunits separated by a vertical distance of two degrees. Thus,
    # the output of a detector cross-multiplies the high-pass component of the corresponding pixel with the low pass 
    # component of the pixel two degrees above the pixel corresponding to the detectors location, and vice-versa, then
    # subtracts the two. Under this formalism, a negative response indicates generally upward motion across the pixel
    # corresponding to the detector, and a positive response indicates downward motion.
    reich[int(2.0/y_res):,:,:] = low_pass[:-int(2.0/y_res),:,:]*high_pass[int(2.0/y_res):,:,:] \
                                    - high_pass[:-int(2.0/y_res),:,:]*low_pass[int(2.0/y_res):,:,:]
        
    
    # Separate the filtered movie into upward and downward components.  
    reich_up = np.copy(reich)
    reich_up[reich_up > 0] = 0
    reich_down = reich
    reich_down[reich_down < 0] = 0
    
    # Zero all pixels which do not correspond to the location of a Reichardt detector.
    reich_up = -reich_up*repmat_3d(reichardt_locs,np.size(movie,2))
    reich_down = reich_down*repmat_3d(reichardt_locs,np.size(movie,2))
    
    return (reich_up,reich_down)
    
    
#                                                 #
#                                                 #
###### default code executed when sim_vs_net ###### 
###### is called without parameters          ######
#                                                 #
#                                                 #
    

if __name__ == "__main__":
    import matplotlib.pyplot as plt

    #enumerate the different types of stimuli
    stim_type = [ 'bar', 'white', 'natural', 'stripe']
    
    #select one stimulus type here by changing the index from 0 to 3
    stim = stim_type[3]
    
    run_time_vars = {'g_gap':1000, # nS
                     'g_inh_scale':0.06, # Factor which multiplies g_gap to give g_inh
                     'g_L_dend':180, # nS
                     'g_L_axon':30, # nS
                     'g_axon_dend':110, #nS
                     'g_exc_in':10000, # nS
                     'g_inh_in':15000, # nS
                     'E_E':60, # mV
                     'E_I':-40, # mV,
                     'tau_m':1.4, # ms
                     'noise_std':6, #10**2.5, # pA
                     'tau_hp':50, # ms
                     'tau_lp':20, # ms
                     'dt_sim':0.01, # ms
                     'dt_im':1, # ms
                     'rot_angle':90,
                     'deg_per_ms': 0.5,
                     'T':100,
                     } 
    
    if stim == 'bar':
        run_time_vars['image'] = {'type':'bar','x_pixels':360,'y_pixels':180,'num_bars':25,'arc_length':40,'arc_width':5}
    elif stim == 'white': 
        run_time_vars['image'] ={'type':'white','x_pixels':360,'y_pixels':180,'upsample':4}
    elif stim == 'natural':                    
        run_time_vars['image'] = {'type':'natural','x_pixels':360,'y_pixels':180,'image_lib_path':'./rotate_image/'}
    elif stim == 'stripe':
        run_time_vars['rot_angle'] = 0
        run_time_vars['deg_per_ms'] = 0.125
        run_time_vars['image'] = {'type':'stripe','x_pixels':360,'y_pixels':180,'strip_rad':5,'freq_s':8}


    (V_L,V_R) = sim_vs_net(run_time_vars)


    T = run_time_vars['T']
    t = np.linspace(0,T,T/run_time_vars['dt_sim'],endpoint=False)
    
    fig1 = plt.figure()
    ax1 = fig1.add_subplot(121)
    ax2 = fig1.add_subplot(122)
    for i in range(10):
        ax1.plot(t,V_L[i,:].T,label='VS'+str(i+1))
        ax2.plot(t,V_R[i,:].T)
    
    ax1.legend()
    ax1.set_title('Left side responses')
    ax2.set_title('Right side responses')
    disp_lim = np.max([np.max(np.abs(V_L)),np.max(np.abs(V_R))])*1.1
    ax1.set_ylim([-disp_lim,disp_lim])
    ax2.set_ylim([-disp_lim,disp_lim])
    
    plt.show()




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