A small neuron-astrocyte network

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This script shows how to create a neuron-astrocyte network in NEST. The network in this script includes 20 neurons and five astrocytes. The astrocytes are modeled with astrocyte_lr_1994, implemented according to [1], [2], and [3]. The neurons are modeled with aeif_cond_alpha_astro, an adaptive exponential integrate-and-fire neuron supporting neuron-astrocyte interactions.

The network is created with the TripartiteConnect() function and the tripartite_bernoulli_with_pool rule (see Tripartite Bernoulli with pool for detailed descriptions). This rule creates a tripartite Bernoulli connectivity with the following principles:

1. For each pair of neurons, a Bernoulli trial with a probability p_primary determines if a tsodyks_synapse will be created between them.

2. For each neuron-neuron connection created, a Bernoulli trial with a probability p_third_if_primary determines if it will be paired with one astrocyte. The selection of this particular astrocyte is confined by pool_size and pool_type (see below).

3. If a neuron-neuron connection is to be paired with an astrocyte, a tsodyks_synapse from the presynaptic (source) neuron to the astrocyte is created, and a sic_connection from the astrocyte to the postsynaptic (target) neuron is created.

The available connectivity parameters are as follows:

• conn_spec parameters

  • p_primary: Connection probability between neurons.

  • p_third_if_primary: Probability of each created neuron-neuron connection to be paired with one astrocyte.

  • pool_size: The size of astrocyte pool for each target neuron. The astrocyte pool of each target neuron is determined before making connections. Each target neuron can only be connected to astrocytes in its pool.

  • pool_type: The way to determine the astrocyte pool for each target neuron. If "random", a number (pool_size) of astrocytes are randomly chosen from all astrocytes (without replacement) and assigned as the pool. If "block", the astrocytes are evenly distributed to the neurons in blocks without overlapping, and the specified pool_size has to be compatible with this arrangement. See Tripartite Bernoulli with pool for more details about pool_type.

• syn_specs parameters

  • primary: syn_spec specifications for the connections between neurons.

  • third_in: syn_spec specifications for the connections from neurons to astrocytes.

  • third_out: syn_spec specifications for the connections from astrocytes to neurons.

In this script, the network is created with the pool_type being "block". p_primary and p_third_if_primary are both set to one to include as many connections as possible. One of the created figures shows the connections between neurons and astrocytes as a result (note that multiple connections may exist between a pair of nodes; this is not obvious in the figure since connections drawn later cover previous ones). It can be seen from the figure that "block" results in astrocytes being connected to postsynaptic neurons in non-overlapping blocks. The pool_size should be compatible with this arrangement; in the case here, a pool_size of one is required. Users can try different parameters (e.g. p_primary = 0.5 and p_third_if_primary = 0.5) to see changes in connections.

With the created network, neuron-astrocyte interactions can be observed. The presynaptic spikes induce the generation of IP3, which then changes the calcium concentration in the astrocytes. This change in calcium then induces the slow inward current (SIC) in the neurons through the sic_connection. The changes in membrane potential of the presynaptic and postsynaptic neurons are also recorded. These data are shown in the created figures.

The pool_type can be changed to “random” to see the results with random astrocyte pools. In that case, the pool_size can be any from one to the total number of astrocytes.

See Tripartite Bernoulli with pool for more details about the TripartiteConnect() function and the tripartite_bernoulli_with_pool rule.

References

[1]

Li, Y. X., & Rinzel, J. (1994). Equations for InsP3 receptor-mediated [Ca2+]i oscillations derived from a detailed kinetic model: a Hodgkin-Huxley like formalism. Journal of theoretical Biology, 166(4), 461-473. DOI: https://doi.org/10.1006/jtbi.1994.1041

[2]

De Young, G. W., & Keizer, J. (1992). A single-pool inositol 1,4,5-trisphosphate-receptor-based model for agonist-stimulated oscillations in Ca2+ concentration. Proceedings of the National Academy of Sciences, 89(20), 9895-9899. DOI: https://doi.org/10.1073/pnas.89.20.9895

[3]

Nadkarni, S., & Jung, P. (2003). Spontaneous oscillations of dressed neurons: a new mechanism for epilepsy?. Physical review letters, 91(26), 268101. DOI: https://doi.org/10.1103/PhysRevLett.91.268101

See Also

Random balanced network with astrocytes

Import all necessary modules.

import matplotlib.pyplot as plt
import nest
import numpy as np

Initialize NEST kernel.

nest.ResetKernel()

Set network parameters.

n_neurons = 10  # number of source and target neurons
n_astrocytes = 5  # number of astrocytes
p_primary = 1.0  # connection probability between neurons
p_third_if_primary = 1.0  # probability of each created neuron-neuron connection to be paired with one astrocyte
pool_size = 1  # astrocyte pool size for each target neuron
pool_type = "block"  # the way to determine the astrocyte pool for each target neuron

Set astrocyte parameters.

astrocyte_model = "astrocyte_lr_1994"
astrocyte_params = {
    "IP3": 0.4,  # IP3 initial value in µM
    "delta_IP3": 2.0,  # parameter determining the increase in astrocytic IP3 concentration induced by synaptic input
    "tau_IP3": 10.0,  # time constant of the exponential decay of astrocytic IP3
}

Set neuron parameters.

neuron_model = "aeif_cond_alpha_astro"
neuron_params = {
    "tau_syn_ex": 2.0,  # excitatory synaptic time constant in ms
    "I_e": 1000.0,  # external current input in pA
}

Functions for plotting.

def plot_connections(conn_n2n, conn_n2a, conn_a2n, pre_id_list, post_id_list, astro_id_list):
    """Plot all connections between neurons and astrocytes.

    Parameters
    ---------
    conn_n2n
        Data of neuron-to-neuron connections.
    conn_n2a
        Data of neuron-to-astrocyte connections.
    conn_a2n
        Data of astrocyte-to-neuron connections.
    pre_id_list
        ID list of presynaptic neurons.
    post_id_list
        ID list of postsynaptic neurons.
    astro_id_list
        ID list of astrocytes.

    """
    print("Plotting connections ...")

    # helper function to create lists of connection positions
    def get_conn_positions(dict_in, source_center, target_center):
        source_list = np.array(dict_in["source"]) - source_center
        target_list = np.array(dict_in["target"]) - target_center
        return source_list.tolist(), target_list.tolist()

    # prepare data (lists of node positions, list of connection positions)
    pre_id_center = (pre_id_list[0] + pre_id_list[-1]) / 2
    post_id_center = (post_id_list[0] + post_id_list[-1]) / 2
    astro_id_center = (astro_id_list[0] + astro_id_list[-1]) / 2
    slist_n2n, tlist_n2n = get_conn_positions(conns_n2n.get(), pre_id_center, post_id_center)
    slist_n2a, tlist_n2a = get_conn_positions(conns_n2a.get(), pre_id_center, astro_id_center)
    slist_a2n, tlist_a2n = get_conn_positions(conns_a2n.get(), astro_id_center, post_id_center)
    # initialize figure
    fig, axs = plt.subplots(1, 1, figsize=(10, 8))
    # plot nodes and connections
    # source neuron nodes
    axs.scatter(
        np.array(pre_id_list) - pre_id_center,
        [2] * len(pre_id_list),
        s=300,
        color="gray",
        marker="^",
        label="pre_neurons",
        zorder=3,
    )
    # neuron-to-neuron connections
    for i, (sx, tx) in enumerate(zip(slist_n2n, tlist_n2n)):
        label = "neuron-to-neuron" if i == 0 else None
        axs.plot([sx, tx], [2, 0], linestyle=":", color="b", alpha=0.3, linewidth=2, label=label)
    # target neuron nodes
    axs.scatter(
        np.array(post_id_list) - post_id_center,
        [0] * len(post_id_list),
        s=300,
        color="k",
        marker="^",
        label="post_neurons",
        zorder=3,
    )
    # neuron-to-astrocyte connections
    for i, (sx, tx) in enumerate(zip(slist_n2a, tlist_n2a)):
        label = "neuron-to-astrocyte" if i == 0 else None
        axs.plot([sx, tx], [2, 1], linestyle="-", color="orange", alpha=0.5, linewidth=2, label=label)
    # astrocyte nodes
    axs.scatter(
        np.array(astro_id_list) - astro_id_center,
        [1] * len(astro_id_list),
        s=300,
        color="g",
        marker="o",
        label="astrocytes",
        zorder=3,
    )
    # astrocyte-to-neuron connections
    for i, (sx, tx) in enumerate(zip(slist_a2n, tlist_a2n)):
        label = "astrocyte-to-neuron" if i == 0 else None
        axs.plot([sx, tx], [1, 0], linestyle="-", color="g", linewidth=4, label=label)
    # set legends
    legend = axs.legend(bbox_to_anchor=(0.5, 1.15), loc="upper center", ncol=3, labelspacing=1.5)
    legend.set_frame_on(False)
    # set axes and frames invisible
    axs.get_xaxis().set_visible(False)
    axs.get_yaxis().set_visible(False)
    axs.spines["top"].set_visible(False)
    axs.spines["bottom"].set_visible(False)
    axs.spines["left"].set_visible(False)
    axs.spines["right"].set_visible(False)


def get_plot_data(data_in, variable):
    """Helper function to get times, means, and standard deviations of data for plotting.

    Parameters
    ---------
    data_in
        Data containing the variable to be plotted.
    Variable
        Variable to be plotted.

    Return values
    -------------
        Times, means, and standard deviations of the variable to be plotted.

    """
    times_all = data_in["times"]
    ts = list(set(data_in["times"]))
    means = np.array([np.mean(data_in[variable][times_all == t]) for t in ts])
    sds = np.array([np.std(data_in[variable][times_all == t]) for t in ts])
    return ts, means, sds


def plot_vm(pre_data, post_data, start):
    """Plot membrane potentials of presynaptic and postsynaptic neurons.

    Parameters
    ---------
    pre_data
        Data of the presynaptic neurons.
    post_data
        Data of the postsynaptic neurons.
    start
       Start time of the data to be plotted.

    """

    print("Plotting V_m ...")
    # get presynaptic data
    pre_times, pre_vm_mean, pre_vm_sd = get_plot_data(pre_data, "V_m")
    # get postsynaptic data
    post_times, post_vm_mean, post_vm_sd = get_plot_data(post_data, "V_m")
    # set plots
    fig, axes = plt.subplots(2, 1, sharex=True)
    color_pre = color_post = "tab:blue"
    # plot presynaptic membrane potential
    axes[0].set_title(f"membrane potential of presynaptic neurons (n={len(set(pre_data['senders']))})")
    axes[0].set_ylabel(r"$V_{m}$ (mV)")
    axes[0].fill_between(
        pre_times, pre_vm_mean + pre_vm_sd, pre_vm_mean - pre_vm_sd, alpha=0.3, linewidth=0.0, color=color_pre
    )
    axes[0].plot(pre_times, pre_vm_mean, linewidth=2, color=color_pre)
    # plot postsynaptic  membrane potential
    axes[1].set_title(f"membrane potential of postsynaptic neurons (n={len(set(post_data['senders']))})")
    axes[1].set_ylabel(r"$V_{m}$ (mV)")
    axes[1].set_xlabel("Time (ms)")
    axes[1].fill_between(
        post_times, post_vm_mean + post_vm_sd, post_vm_mean - post_vm_sd, alpha=0.3, linewidth=0.0, color=color_post
    )
    axes[1].plot(post_times, post_vm_mean, linewidth=2, color=color_post)


def plot_dynamics(astro_data, neuron_data, start):
    """Plot dynamics in astrocytes and SIC in neurons.

    Parameters
    ---------
    astro_data
        Data of the astrocytes.
    neuron_data
        Data of the neurons.
    start
       Start time of the data to be plotted.

    """
    print("Plotting dynamics ...")
    # get astrocyte data
    astro_times, astro_ip3_mean, astro_ip3_sd = get_plot_data(astro_data, "IP3")
    astro_times, astro_ca_mean, astro_ca_sd = get_plot_data(astro_data, "Ca_astro")
    # get neuron data
    neuron_times, neuron_sic_mean, neuron_sic_sd = get_plot_data(neuron_data, "I_SIC")
    # set plots
    fig, axes = plt.subplots(2, 1, sharex=True)
    color_ip3 = "tab:blue"
    color_cal = "tab:green"
    color_sic = "tab:purple"
    # plot astrocyte data
    n_astro = len(set(astro_data["senders"]))
    axes[0].set_title(f"IP$_{{3}}$ and Ca$^{{2+}}$ in astrocytes (n={n_astro})")
    axes[0].set_ylabel(r"IP$_{3}$ ($\mu$M)")
    axes[0].tick_params(axis="y", labelcolor=color_ip3)
    axes[0].fill_between(
        astro_times,
        astro_ip3_mean + astro_ip3_sd,
        astro_ip3_mean - astro_ip3_sd,
        alpha=0.3,
        linewidth=0.0,
        color=color_ip3,
    )
    axes[0].plot(astro_times, astro_ip3_mean, linewidth=2, color=color_ip3)
    ax = axes[0].twinx()
    ax.set_ylabel(r"Ca$^{2+}$ ($\mu$M)")
    ax.tick_params(axis="y", labelcolor=color_cal)
    ax.fill_between(
        astro_times, astro_ca_mean + astro_ca_sd, astro_ca_mean - astro_ca_sd, alpha=0.3, linewidth=0.0, color=color_cal
    )
    ax.plot(astro_times, astro_ca_mean, linewidth=2, color=color_cal)
    # plot neuron data
    n_neuron = len(set(neuron_data["senders"]))
    axes[1].set_title(f"SIC in postsynaptic neurons (n={n_neuron})")
    axes[1].set_ylabel("SIC (pA)")
    axes[1].set_xlabel("Time (ms)")
    axes[1].fill_between(
        neuron_times,
        neuron_sic_mean + neuron_sic_sd,
        neuron_sic_mean - neuron_sic_sd,
        alpha=0.3,
        linewidth=0.0,
        color=color_sic,
    )
    axes[1].plot(neuron_times, neuron_sic_mean, linewidth=2, color=color_sic)

Create and connect populations and devices. The neurons and astrocytes are connected with multimeters to record their dynamics.

pre_neurons = nest.Create(neuron_model, n_neurons, params=neuron_params)
post_neurons = nest.Create(neuron_model, n_neurons, params=neuron_params)
astrocytes = nest.Create(astrocyte_model, n_astrocytes, params=astrocyte_params)
nest.TripartiteConnect(
    pre_neurons,
    post_neurons,
    astrocytes,
    conn_spec={
        "rule": "tripartite_bernoulli_with_pool",
        "p_primary": p_primary,
        "p_third_if_primary": p_third_if_primary,
        "pool_size": pool_size,
        "pool_type": pool_type,
    },
    syn_specs={
        "primary": {"synapse_model": "tsodyks_synapse"},
        "third_in": {"synapse_model": "tsodyks_synapse"},
        "third_out": {"synapse_model": "sic_connection"},
    },
)
mm_pre_neurons = nest.Create("multimeter", params={"record_from": ["V_m"]})
mm_post_neurons = nest.Create("multimeter", params={"record_from": ["V_m", "I_SIC"]})
mm_astrocytes = nest.Create("multimeter", params={"record_from": ["IP3", "Ca_astro"]})
nest.Connect(mm_pre_neurons, pre_neurons)
nest.Connect(mm_post_neurons, post_neurons)
nest.Connect(mm_astrocytes, astrocytes)

Get connection data. The data are used to plot the network connectivity.

conns_a2n = nest.GetConnections(astrocytes, post_neurons)
conns_n2n = nest.GetConnections(pre_neurons, post_neurons)
conns_n2a = nest.GetConnections(pre_neurons, astrocytes)

Run simulation.

nest.Simulate(1000.0)
plot_connections(conns_n2n, conns_n2a, conns_a2n, pre_neurons.tolist(), post_neurons.tolist(), astrocytes.tolist())
plot_vm(mm_pre_neurons.events, mm_post_neurons.events, 0.0)
plot_dynamics(mm_astrocytes.events, mm_post_neurons.events, 0.0)
plt.show()