.. currentmodule:: brian2 .. example_3_io_synapse: Example: example_3_io_synapse ============================= .. only:: html .. |launchbinder| image:: http://mybinder.org/badge.svg .. _launchbinder: https://mybinder.org/v2/gh/brian-team/brian2-binder/master?filepath=examples/frompapers/Stimberg_et_al_2018/example_3_io_synapse.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 3: Modeling of modulation of synaptic release by gliotransmission. Three synapses: the first one without astrocyte, the remaining two respectively with open-loop and close-loop gliotransmission (see De Pitta' et al., 2011, 2016) :: from brian2 import * import plot_utils as pu set_device('cpp_standalone', directory=None) # Use fast "C++ standalone mode" ################################################################################ # Model parameters ################################################################################ ### General parameters transient = 16.5*second duration = transient + 600*ms # Total simulation time sim_dt = 1*ms # Integrator/sampling step ### 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 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 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" # We are only interested in the activity of the synapse, so we replace the # neurons by trivial "dummy" groups spikes = [0, 50, 100, 150, 200, 300, 310, 320, 330, 340, 350, 360, 370, 380, 390, 400]*ms spikes += transient # allow for some initial transient source_neurons = SpikeGeneratorGroup(1, np.zeros(len(spikes)), spikes) target_neurons = NeuronGroup(1, '') ### 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 (clock-driven) # Fraction of synaptic neurotransmitter resources available: dx_S/dt = Omega_d *(1 - x_S) : 1 (clock-driven) # released synaptic neurotransmitter resources: r_S : 1 # gliotransmitter concentration in the extracellular space: G_A : mmolar ''' 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, only the second and third ones are modulated by astrocytes synapses.connect(True, n=3) ### Astrocytes # The astrocyte emits gliotransmitter when its Ca^2+ concentration crosses # a threshold astro_eqs = ''' # IP_3 dynamics: dI/dt = J_delta - J_3K - J_5P + J_ex : mmolar 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 # Exogenous stimulation 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: dx_A/dt = Omega_A * (1 - x_A) : 1 # gliotransmitter concentration in the extracellular space: dG_A/dt = -Omega_e*G_A : mmolar ''' glio_release = ''' G_A += rho_e * G_T * U_A * x_A x_A -= U_A * x_A ''' # The following formulation makes sure that a "spike" is only triggered at the # first threshold crossing -- the astrocyte is considered "refractory" (i.e., # not allowed to trigger another event) as long as the Ca2+ concentration # remains above threshold # The gliotransmitter release happens when the threshold is crossed, in Brian # terms it can therefore be considered a "reset" astrocyte = NeuronGroup(2, astro_eqs, threshold='C>C_Theta', refractory='C>C_Theta', reset=glio_release, method='rk4') # Different length of stimulation astrocyte.x_A = 1.0 astrocyte.h = 0.9 astrocyte.I = 0.4*umolar astrocyte.I_bias = np.asarray([0.8, 1.25])*umolar # Connection between astrocytes and the second synapse. Note that in this # special case, where the synapse is only influenced by the gliotransmitter from # a single astrocyte, the '(linked)' variable mechanism could be used instead. # The mechanism used below is more general and can add the contribution of # several astrocytes ecs_astro_to_syn = Synapses(astrocyte, synapses, 'G_A_post = G_A_pre : mmolar (summed)') # Connect second and third synapse to a different astrocyte ecs_astro_to_syn.connect(j='i+1') ################################################################################ # Monitors ################################################################################ # Note that we cannot use "record=True" for synapses in C++ standalone mode -- # the StateMonitor needs to know the number of elements to record from during # its initialization, but in C++ standalone mode, no synapses have been created # yet. We therefore explicitly state to record from the three synapses. syn_mon = StateMonitor(synapses, variables=['u_S', 'x_S', 'r_S', 'Y_S'], record=[0, 1, 2]) ast_mon = StateMonitor(astrocyte, variables=['C', 'G_A'], record=True) ################################################################################ # Simulation run ################################################################################ run(duration, report='text') ################################################################################ # Analysis and plotting ################################################################################ from matplotlib import cycler plt.style.use('figures.mplstyle') fig, ax = plt.subplots(nrows=7, ncols=1, figsize=(6.26894, 6.26894 * 1.2), gridspec_kw={'height_ratios': [3, 2, 1, 1, 3, 3, 3], 'top': 0.98, 'bottom': 0.08, 'left': 0.15, 'right': 0.95}) ## Ca^2+ traces of the two astrocytes ax[0].plot((ast_mon.t-transient)/second, ast_mon.C[0]/umolar, '-', color='C2') ax[0].plot((ast_mon.t-transient)/second, ast_mon.C[1]/umolar, '-', color='C3') ## Add threshold for gliotransmitter release ax[0].plot(np.asarray([-transient/second, 0.0]), np.asarray([C_Theta, C_Theta])/umolar, ':', color='gray') ax[0].set(xlim=[-transient/second, 0.0], yticks=[0., 0.4, 0.8, 1.2], ylabel=r'$C$ ($\mu$M)') pu.adjust_spines(ax[0], ['left']) ## Gliotransmitter concentration in the extracellular space ax[1].plot((ast_mon.t-transient)/second, ast_mon.G_A[0]/umolar, '-', color='C2') ax[1].plot((ast_mon.t-transient)/second, ast_mon.G_A[1]/umolar, '-', color='C3') ax[1].set(yticks=[0., 50., 100.], xlim=[-transient/second, 0.0], xlabel='time (s)', ylabel=r'$G_A$ ($\mu$M)') pu.adjust_spines(ax[1], ['left', 'bottom']) ## Turn off one axis to display x-labeling of ax[1] correctly ax[2].axis('off') ## Synaptic stimulation ax[3].vlines((spikes-transient)/ms, 0, 1, clip_on=False) ax[3].set(xlim=(0, (duration-transient)/ms)) ax[3].axis('off') ## Synaptic variables # Use a custom cycle that uses black as the first color prop_cycle = cycler(color='k').concat(matplotlib.rcParams['axes.prop_cycle'][2:]) ax[4].set(xlim=(0, (duration-transient)/ms), ylim=[0., 1.], yticks=np.arange(0, 1.1, .25), ylabel='$u_S$', prop_cycle=prop_cycle) ax[4].plot((syn_mon.t-transient)/ms, syn_mon.u_S.T) pu.adjust_spines(ax[4], ['left']) ax[5].set(xlim=(0, (duration-transient)/ms), ylim=[-0.05, 1.], yticks=np.arange(0, 1.1, .25), ylabel='$x_S$', prop_cycle=prop_cycle) ax[5].plot((syn_mon.t-transient)/ms, syn_mon.x_S.T) pu.adjust_spines(ax[5], ['left']) ax[6].set(xlim=(0, (duration-transient)/ms), ylim=(-5., 1500), xticks=np.arange(0, (duration-transient)/ms, 100), xlabel='time (ms)', yticks=[0, 500, 1000, 1500], ylabel=r'$Y_S$ ($\mu$M)', prop_cycle=prop_cycle) ax[6].plot((syn_mon.t-transient)/ms, syn_mon.Y_S.T/umolar) ax[6].legend(['no gliotransmission', 'weak gliotransmission', 'stronger gliotransmission'], loc='upper right') pu.adjust_spines(ax[6], ['left', 'bottom']) pu.adjust_ylabels(ax, x_offset=-0.11) plt.show() .. image:: ../resources/examples_images/frompapers.Stimberg_et_al_2018.example_3_io_synapse.1.png