.. currentmodule:: brian2

.. example_4_synrel:

Example: example_4_synrel
=========================


        .. only:: html

            .. |launchbinder| image:: file:///usr/share/doc/python-brian-doc/docs/badge.svg
            .. _launchbinder: https://mybinder.org/v2/gh/brian-team/brian2-binder/master?filepath=examples/frompapers/Stimberg_et_al_2018/example_4_synrel.ipynb

            .. note::
               You can launch an interactive, editable version of this
               example without installing any local files
               using the Binder service (although note that at some times this
               may be slow or fail to open): |launchbinder|_

        

Modeling neuron-glia interactions with the Brian 2 simulator
Marcel Stimberg, Dan F. M. Goodman, Romain Brette, Maurizio De Pittà
bioRxiv 198366; doi: https://doi.org/10.1101/198366

Figure 4B: Closed-loop gliotransmission.

Extracellular neurotransmitter concentration (averaged across 500 synapses) for three
step increases of the presynaptic rate, for three synapses: one without gliotransmission,
and the other two with open- and close-loop gliotransmssion.

::

    from brian2 import *
    
    import plot_utils as pu
    
    set_device('cpp_standalone', directory=None)  # Use fast "C++ standalone mode"
    seed(16283)  # to get identical figures for repeated runs
    
    ################################################################################
    # Model parameters
    ################################################################################
    ### General parameters
    N_synapses = 500
    N_astro = 2
    duration = 20*second         # Total simulation time
    sim_dt = 1*ms                # Integrator/sampling step
    
    ### Neuron parameters
    
    # ### Synapse parameters
    ### Synapse parameters
    rho_c = 0.005                # Synaptic vesicle-to-extracellular space volume ratio
    Y_T = 500*mmolar             # Total vesicular neurotransmitter concentration
    Omega_c = 40/second          # Neurotransmitter clearance rate
    U_0__star = 0.6              # Resting synaptic release probability
    Omega_f = 3.33/second        # Synaptic facilitation rate
    Omega_d = 2.0/second         # Synaptic depression rate
    # --- Presynaptic receptors
    O_G = 1.5/umolar/second      # Agonist binding (activating) rate
    Omega_G = 0.5/(60*second)    # Agonist release (deactivating) rate
    
    ### Astrocyte parameters
    # ---  Calcium fluxes
    O_P = 0.9*umolar/second      # Maximal Ca^2+ uptake rate by SERCAs
    K_P = 0.05 * umolar          # Ca2+ affinity of SERCAs
    C_T = 2*umolar               # Total cell free Ca^2+ content
    rho_A = 0.18                 # ER-to-cytoplasm volume ratio
    Omega_C = 6/second           # Maximal rate of Ca^2+ release by IP_3Rs
    Omega_L = 0.1/second         # Maximal rate of Ca^2+ leak from the ER
    # --- IP_3R kinectics
    d_1 = 0.13*umolar            # IP_3 binding affinity
    d_2 = 1.05*umolar            # Ca^2+ inactivation dissociation constant
    O_2 = 0.2/umolar/second      # IP_3R binding rate for Ca^2+ inhibition
    d_3 = 0.9434*umolar          # IP_3 dissociation constant
    d_5 = 0.08*umolar            # Ca^2+ activation dissociation constant
    # --- IP_3 production
    # --- Agonist-dependent IP_3 production
    O_beta = 3.2*umolar/second   # Maximal rate of IP_3 production by PLCbeta
    O_N = 0.3/umolar/second      # Agonist binding rate
    Omega_N = 0.5/second         # Maximal inactivation rate
    K_KC = 0.5*umolar            # Ca^2+ affinity of PKC
    zeta = 10                    # Maximal reduction of receptor affinity by PKC
    # --- Endogenous IP3 production
    O_delta = 0.6*umolar/second  # Maximal rate of IP_3 production by PLCdelta
    kappa_delta = 1.5* umolar    # Inhibition constant of PLC_delta by IP_3
    K_delta = 0.1*umolar         # Ca^2+ affinity of PLCdelta
    # --- IP_3 diffusion
    F = 2*umolar/second          # GJC IP_3 permeability
    I_Theta = 0.3*umolar         # Threshold gradient for IP_3 diffusion
    omega_I = 0.05*umolar        # Scaling factor of diffusion
    # --- IP_3 degradation
    Omega_5P = 0.05/second       # Maximal rate of IP_3 degradation by IP-5P
    K_D = 0.7*umolar             # Ca^2+ affinity of IP3-3K
    K_3K = 1.0*umolar            # IP_3 affinity of IP_3-3K
    O_3K = 4.5*umolar/second     # Maximal rate of IP_3 degradation by IP_3-3K
    # --- IP_3 diffusion
    F_ex = 2.0*umolar/second     # Maximal exogenous IP3 flow
    I_Theta = 0.3*umolar         # Threshold gradient for IP_3 diffusion
    omega_I = 0.05*umolar        # Scaling factor of diffusion
    # --- Gliotransmitter release and time course
    C_Theta = 0.5*umolar         # Ca^2+ threshold for exocytosis
    Omega_A = 0.6/second         # Gliotransmitter recycling rate
    U_A = 0.6                    # Gliotransmitter release probability
    G_T = 200*mmolar             # Total vesicular gliotransmitter concentration
    rho_e = 6.5e-4               # Astrocytic vesicle-to-extracellular volume ratio
    Omega_e = 60/second          # Gliotransmitter clearance rate
    alpha = 0.0                  # Gliotransmission nature
    
    ################################################################################
    # Model definition
    ################################################################################
    defaultclock.dt = sim_dt  # Set the integration time
    
    # ### "Neurons"
    rate_in = TimedArray([0.011, 0.11, 1.1, 11] * Hz, dt=5*second)
    source_neurons = PoissonGroup(N_synapses, rates='rate_in(t)')
    target_neurons = NeuronGroup(N_synapses, '')
    
    ### Synapses
    # Note that the synapse does not actually have any effect on the post-synaptic
    # target
    # Also note that for easier plotting we do not use the "event-driven" flag here,
    # even though the value of u_S and x_S only needs to be updated on the arrival
    # of a spike
    synapses_eqs = '''
    # Neurotransmitter
    dY_S/dt = -Omega_c * Y_S : mmolar (clock-driven)
    # Fraction of activated presynaptic receptors
    dGamma_S/dt = O_G * G_A * (1 - Gamma_S) - Omega_G * Gamma_S : 1 (clock-driven)
    # Usage of releasable neurotransmitter per single action potential:
    du_S/dt = -Omega_f * u_S : 1 (event-driven)
    # Fraction of synaptic neurotransmitter resources available for release:
    dx_S/dt = Omega_d *(1 - x_S) : 1 (event-driven)
    r_S : 1  # released synaptic neurotransmitter resources
    G_A : mmolar  # gliotransmitter concentration in the extracellular space
    '''
    synapses_action = '''
    U_0 = (1 - Gamma_S) * U_0__star + alpha * Gamma_S
    u_S += U_0 * (1 - u_S)
    r_S = u_S * x_S
    x_S -= r_S
    Y_S += rho_c * Y_T * r_S
    '''
    synapses = Synapses(source_neurons, target_neurons,
                        model=synapses_eqs, on_pre=synapses_action,
                        method='exact')
    # We create three synapses per connection: only the first two are modulated by
    # the astrocyte however. Note that we could also create three synapses per
    # connection with a single connect call by using connect(j='i', n=3), but this
    # would create synapses arranged differently (synapses connection pairs
    # (0, 0), (0, 0), (0, 0), (1, 1), (1, 1), (1, 1), ..., instead of
    # connections (0, 0), (1, 1), ..., (0, 0), (1, 1), ..., (0, 0), (1, 1), ...)
    # making the later connection descriptions more complicated.
    synapses.connect(j='i')  # closed-loop modulation
    synapses.connect(j='i')  # open modulation
    synapses.connect(j='i')  # no modulation
    synapses.x_S = 1.0
    
    ### Astrocytes
    # The astrocyte emits gliotransmitter when its Ca^2+ concentration crosses
    # a threshold
    astro_eqs = '''
    # Fraction of activated astrocyte receptors:
    dGamma_A/dt = O_N * Y_S * (1 - Gamma_A) -
                  Omega_N*(1 + zeta * C/(C + K_KC)) * Gamma_A : 1
    
    # IP_3 dynamics:
    dI/dt = J_beta + J_delta - J_3K - J_5P + J_ex             : mmolar
    J_beta = O_beta * Gamma_A                                 : mmolar/second
    J_delta = O_delta/(1 + I/kappa_delta) *
                             C**2/(C**2 + K_delta**2)         : mmolar/second
    J_3K = O_3K * C**4/(C**4 + K_D**4) * I/(I + K_3K)         : mmolar/second
    J_5P = Omega_5P*I                                         : mmolar/second
    delta_I_bias = I - I_bias : mmolar
    J_ex = -F_ex/2*(1 + tanh((abs(delta_I_bias) - I_Theta)/omega_I)) *
                    sign(delta_I_bias)                        : mmolar/second
    I_bias                                                    : mmolar (constant)
    
    # Ca^2+-induced Ca^2+ release:
    dC/dt = (Omega_C * m_inf**3 * h**3 + Omega_L) * (C_T - (1 + rho_A)*C) -
            O_P * C**2/(C**2 + K_P**2) : mmolar
    dh/dt = (h_inf - h)/tau_h          : 1  # IP3R de-inactivation probability
    m_inf = I/(I + d_1) * C/(C + d_5)  : 1
    h_inf = Q_2/(Q_2 + C)              : 1
    tau_h = 1/(O_2 * (Q_2 + C))        : second
    Q_2 = d_2 * (I + d_1)/(I + d_3)    : mmolar
    
    # Fraction of gliotransmitter resources available for release
    dx_A/dt = Omega_A * (1 - x_A) : 1
    # gliotransmitter concentration in the extracellular space
    dG_A/dt = -Omega_e*G_A        : mmolar
    # Neurotransmitter concentration in the extracellular space
    Y_S                           : mmolar
    '''
    glio_release = '''
    G_A += rho_e * G_T * U_A * x_A
    x_A -= U_A *  x_A
    '''
    astrocyte = NeuronGroup(N_astro*N_synapses, astro_eqs,
                            # The following formulation makes sure that a "spike" is
                            # only triggered at the first threshold crossing
                            threshold='C>C_Theta',
                            refractory='C>C_Theta',
                            # The gliotransmitter release happens when the threshold
                            # is crossed, in Brian terms it can therefore be
                            # considered a "reset"
                            reset=glio_release,
                            method='rk4')
    astrocyte.h = 0.9
    astrocyte.x_A = 1.0
    # Only the second group of N_synapses astrocytes are activated by external stimulation
    astrocyte.I_bias = (np.r_[np.zeros(N_synapses), np.ones(N_synapses)])*1.0*umolar
    
    ## Connections
    ecs_syn_to_astro = Synapses(synapses, astrocyte,
                                'Y_S_post = Y_S_pre : mmolar (summed)')
    # Connect the first N_synapses synapses--astrocyte pairs
    ecs_syn_to_astro.connect(j='i if i < N_synapses')
    ecs_astro_to_syn = Synapses(astrocyte, synapses,
                                'G_A_post = G_A_pre : mmolar (summed)')
    # Connect the first N_synapses astrocytes--pairs (closed-loop)
    ecs_astro_to_syn.connect(j='i if i < N_synapses')
    # Connect the second N_synapses astrocyte--synapses pairs (open-loop)
    ecs_astro_to_syn.connect(j='i if i >= N_synapses and i < 2*N_synapses')
    
    ################################################################################
    # Monitors
    ################################################################################
    syn_mon = StateMonitor(synapses, 'Y_S',
                           record=np.arange(N_synapses*(N_astro+1)), dt=10*ms)
    
    ################################################################################
    # Simulation run
    ################################################################################
    run(duration, report='text')
    
    ################################################################################
    # Analysis and plotting
    ################################################################################
    plt.style.use('figures.mplstyle')
    
    fig, ax = plt.subplots(nrows=4, ncols=1, figsize=(3.07, 3.07*1.33),
                           sharex=False,
                           gridspec_kw={'height_ratios': [1, 3, 3, 3],
                                        'top': 0.98, 'bottom': 0.12,
                                        'left': 0.24, 'right': 0.95})
    ax[0].semilogy(syn_mon.t/second, rate_in(syn_mon.t), '-', color='black')
    ax[0].set(xlim=(0, duration/second), ylim=(0.01, 12),
              yticks=[0.01, 0.1, 1, 10], ylabel=r'$\nu_{in}$ (Hz)')
    ax[0].yaxis.set_major_formatter(ScalarFormatter())
    pu.adjust_spines(ax[0], ['left'])
    
    ax[1].plot(syn_mon.t/second,
               np.mean(syn_mon.Y_S[2*N_synapses:]/umolar, axis=0),
               '-', color='black')
    ax[1].set(xlim=(0, duration/second), ylim=(-5, 260),
              yticks=np.arange(0, 260, 50),
              ylabel=r'$\langle Y_S \rangle$ ($\mu$M)')
    ax[1].legend(['no gliotransmission'], loc='upper left')
    pu.adjust_spines(ax[1], ['left'])
    
    ax[2].plot(syn_mon.t/second,
               np.mean(syn_mon.Y_S[N_synapses:2*N_synapses]/umolar, axis=0),
               '-', color='C2')
    ax[2].set(xlim=(0, duration/second), ylim=(-3, 150),
              yticks=np.arange(0, 151, 25),
              ylabel=r'$\langle Y_S \rangle$ ($\mu$M)')
    ax[2].legend(['open-loop gliotransmission'], loc='upper left')
    pu.adjust_spines(ax[2], ['left'])
    
    ax[3].plot(syn_mon.t/second,
               np.mean(syn_mon.Y_S[:N_synapses]/umolar, axis=0),
               '-', color='C3')
    ax[3].set(xlim=(0, duration/second), ylim=(-2, 150),
              xticks=np.arange(0., duration/second+1, 5.0),
              yticks=np.arange(0, 151, 25),
              xlabel='time (s)', ylabel=r'$\langle Y_S \rangle$ ($\mu$M)')
    ax[3].legend(['closed-loop gliotransmission'], loc='upper left')
    pu.adjust_spines(ax[3], ['left', 'bottom'])
    
    pu.adjust_ylabels(ax, x_offset=-0.22)
    
    plt.show()

