# Spiking Neuron-Astrocyte Networks for Image Recognition

Jhunlyn Lorenzo, Juan-Antonio Rico-Gallego, Stéphane Binczak, Sabir Jacquir

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Jhunlyn Lorenzo, Juan-Antonio Rico-Gallego, Stéphane Binczak, Sabir Jacquir. Spiking Neuron-Astrocyte Networks for Image Recognition. Neural Computation, 2025, 37 (4), pp.635-665. 10.1162/neco_a_01740. hal-04982224

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# Spiking Neuron-Astrocyte Networks for Image Recognition

**Jhunlyn Lorenzo**

Jhunlyn.Lorenzo@u-bourgogne.fr
Laboratory ImViA EA7535, Université de Bourgogne, 21078 Dijon, France, and
College of Engineering and Information Technology, Cavite State University, 4122,
Indang, Philippines

**Juan-Antonio Rico-Gallego**

juanantonio.rico@computaex.es
Foundation for Computing and Advanced Technologies of Extremadura,
Extremadura Supercomputing Center, 10071, Cáceres

**Stéphane Binczak**

stbinc@u-bourgogne.fr
Laboratory ImViA EA7535, Université de Bourgogne,
21078 Dijon, France

**Sabir Jacquir**

sabir.jacquir@universite-paris-saclay.fr
Université Paris-Saclay, CNRS, Institut des Neurosciences Paris-Saclay,
91190 Gif-sur-Yvette, France

From biological and artificial network perspectives, researchers have started acknowledging astrocytes as computational units mediating neural processes. Here, we propose a novel biologically inspired neuron-astrocyte network model for image recognition, one of the first attempts at implementing astrocytes in spiking neuron networks (SNNs) using a standard data set. The architecture for image recognition has three primary units: the preprocessing unit for converting the image pixels into spiking patterns, the neuron-astrocyte network forming bipartite (neural connections) and tripartite synapses (neural and astrocytic connections), and the classifier unit. In the astrocyte-mediated SNNs, an astrocyte integrates neural signals following the simplified Postnov model. It then modulates the integrate-and-fire (IF) neurons via gliotransmission, thereby strengthening the synaptic connections of the neurons within the astrocytic territory. We develop an architecture derived from a baseline SNN model for unsupervised digit classification. The spiking neuron-astrocyte networks (SNANs) display better network performance

<footnote>Jhunlyn Lorenzo is the corresponding author.</footnote>
with an optimal variance-bias trade-off than SNN alone. We demonstrate that astrocytes promote faster learning, support memory formation and recognition, and provide a simplified network architecture. Our proposed SNAN can serve as a benchmark for future researchers on astrocyte implementation in artificial networks, particularly in neuromorphic systems, for its simplified design.

# 1 Introduction

Twenty years ago, Araque et al. (1999) introduced the concept of the tripartite synapse, where the astrocyte acts as the third component in synaptic communication (Araque et al., 1999). An astrocyte, through its peripheral processes, enwraps the synaptic space between the pre- and postsynaptic neurons, as shown in Figure 1. The presynaptic neuron encodes information in the action potential (AP) pattern (in the spiking activity of its membrane potential) and sends it to the neighboring neurons through the neurotransmission process, wherein glutamate molecules ($Glu^−$) are released into the synaptic space (Südhof, 2012). A fraction of synaptic $Glu^−$ activates the postsynaptic $Glu^−$ receptors ($GluRs$) or diffuses to the extrasynaptic space and activates the metabotropic $GluRs$ ($mGluRs$) on the astrocytic processes (Nedergaard & Verkhratsky, 2012). Therefore, the presynaptic neuron also modulates the activity of the astrocyte by indirectly producing inositol triphosphate ($IP_3$) molecules that result in the increase of calcium concentration ($[Ca^{2+}]$) in the intracellular space (Volterra & Meldolesi, 2005). When the intracellular $[Ca^{2+}]$ increases above a threshold ($Ca^{2+}$ spike), the astrocyte releases extrasynaptic $Glu^−$. In this case, the astrocyte modulates the postsynaptic neuronal components by activating extrasynaptic $GluRs$ (Wu et al., 2014). The postsynaptic neuron integrates the fast and slow excitatory signals produced by neuron- and gliotransmission, respectively. Therefore, the firing patterns of the presynaptic neuron and the astrocytic process work in conjunction to regulate postsynaptic signals. The tripartite synapse concept implies that astrocytes not only provide structural support to neurons but also serve as pivotal elements in neuronal communication (Savtchouk & Volterra, 2018).

Extending the scope of investigation beyond the tripartite synapse to encompass the entirety of the astrocytic cell reveals that a single astrocyte can enwrap numerous synapses originating from multiple pre- and postsynaptic neurons (Zhou et al., 2019), as illustrated in Figure 2. $Ca^{2+}$ signaling, which is primarily triggered by $IP_3$ production, is the main signaling mechanism of astrocytes (Haydon & Nedergaard, 2015). The astrocytic processes can be viewed as distinct compartments, each having independent $Ca^{2+}$ signaling mechanisms. These compartments can communicate with one another via the intracellular diffusion of $Ca^{2+}$ and $IP_3$ into the soma or adjacent compartments, and vice versa (Bindocci et al., 2017; Covelo &
<figure><img src="image_2.png" /></figure>

Figure 1: The tripartite synapse. The closed-loop dynamics of the tripartite synapse begin with presynaptic neuron activation via action potentials, releasing neurotransmitters that bind to both postsynaptic and astrocytic Glu⁻ receptors. Astrocytic receptor binding triggers IP₃ production, raising intracellular Ca²⁺ levels. When [Ca²⁺] surpasses a threshold, the astrocyte releases gliotransmitters into the extracellular space that activate extrasynaptic Glu⁻ receptors on the postsynaptic neuron. This action modulates synaptic activity and contributes to the postsynaptic potential, enhancing overall synaptic communication through this astrocyte-mediated loop.

Araque, 2016). Given the complex architecture of astrocytes, there exist different pathways by which astrocytes can modulate neighboring neurons. For example, a tripartite synapse that remains mostly inactive due to the absence of presynaptic activity can be activated by Ca²⁺ that diffuses from both the astrocytic soma and other active compartments (Oberheim et al., 2012). All tripartite synapses also can be activated synchronously during a global Ca²⁺ elevation caused by the integration of Ca²⁺ from all compartments (Guerra-Gomes et al., 2018).

Astrocytes are key players in synapse formation (Pastur-Romay et al., 2016) and in enhancing neuronal synchrony (Pankratova et al., 2019) by regulating stochastic neuronal spiking. Another advantage of astrocytes in the neural network is their function in memory storage—in maintaining long-term hippocampal potentiation (Min et al., 2012). However, the use of astrocytes as components in artificial neural networks is a more recent concept. Why should astrocytes be considered as computational units in neural networks? Analogous to the McCulloch-Pitts neuron model (McCulloch & Pitts, 1943), which was inspired by biological neurons and laid the foundation for modern artificial neural networks, astrocytes demonstrate several similarities to neurons despite their relatively slower
<figure><img src="image_3.png" /></figure>

Figure 2: An astrocyte forming tripartite synapses. Calcium and $IP_3$ molecules diffuse bidirectionally between astrocytic processes and the soma, with each acting as a separate compartment with distinct dynamics. This compartmentalization allows astrocytes to integrate signals from nearby synapses and modulate local synaptic environments individually. The diffusion across compartments also coordinates astrocytic responses, enabling synchronized signaling and broader network modulation.

signaling dynamics. First, both neurons and astrocytes can modulate synaptic weights with neurons in adjacent layers via neurotransmission and gliotransmission (Bindocci et al., 2017). Second, neurons integrate synaptic inputs and generate spikes upon reaching a specific threshold, whereas astrocytes integrate $Ca^{2+}$ signals and spike once the $Ca^{2+}$ threshold has been attained (Nedergaard & Verkhratsky, 2012). And third, astrocytes also form astrocytic networks in which each unit communicates via $Ca^{2+}$ wave propagation (Wallach et al., 2014). Previous research showed that the addition of astrocytes into the neuronal network is advantageous in terms of significantly improving neural communication and synchrony (Wade et al., 2011) and pattern separation (Hassanpoor & Saidi, 2020).

In the context of artificial neural networks (ANNs), integrating artificial astrocytes into the network architecture has been shown to enhance classification performance, as demonstrated by Porto-Pazos et al. (2011) and by Han and colleagues with their AstroNet (Han et al., 2023) and MA-Net (Han et al., 2024) models. As the field progresses, the third generation of neural networks, known as spiking neural networks (SNNs), is becoming increasingly competitive with their ANN counterparts. This is primarily due to their ability to emulate brain-like dynamics and perform event-based
computations (Lee et al., 2020; Lobo et al., 2020; Pfeiffer & Pfeil, 2018). Inspired by biological neural mechanisms, SNNs employ spike-based signaling to perform brain-like computations, integrate and represent spatiotemporal information, process sparse and asynchronous signals, and execute parallel processing (Wang et al., 2020). Similar to actual neural circuits implemented in neuromorphic circuits, SNNs offer low-power consumption, analog computation, rapid inference, online learning, and event-driven processing. The temporal dimensions of SNNs, in particular, make them prime candidates for real-world applications and neuromorphic hardware implementations compared to other deep neural networks (Farsa et al., 2019).

While models of SNNs closely mimic brain processes, there are some drawbacks. Training using benchmarks such as MNIST (Modified National Institute of Standards and Technology; Deng, 2012) and ImageNet (Deng et al., 2009) indeed yields a lower accuracy than ANNs, which can be attributed to converting the frame-based images into rate-coded information. Besides, there is a lack of training algorithms for spiking networks. Training SNNs means dealing with the asynchronous and discontinuous nature of spikes, making the application of current differentiable backpropagation techniques quite challenging (Farsa et al., 2019; Pfeiffer & Pfeil, 2018). Here, we draw inspiration from biological processes to address these issues, particularly from the astrocyte-mediated synapse—the tripartite synapse. This study is a proof of concept that astrocytes can be employed in image recognition to improve SNNs’ performance. While recent examples include the bio-inspired neuron-astrocyte network for image classification (Nazari et al., 2019; Nazari & Faez, 2019), noise suppression in dynamic images (Stasenko & Kazantsev, 2023), and the unsupervised network with self-repair in the event of synaptic failure (Naghieh et al., 2023; Rastogi et al., 2021), astrocytes employed in these examples are for neural support rather than as computational units (Kastanenka et al., 2020).

This study presents a novel approach for improving image recognition performance through the integration of astrocytes in spiking neural networks. The proposed spiking neural network with astrocyte network (SNAN) architecture includes simplified neuron and astrocyte models, allowing for efficient and accurate information processing, and employs an unsupervised learning scheme based on spike-timing-dependent plasticity (STDP). Subsequently, the network was trained using the MNIST data set to determine the optimal hyperparameters concerning network architecture and learning. We optimized network parameters such as synaptic coupling strengths and firing thresholds to enhance network performance. Upon training the network, we evaluate its accuracy in predicting the output digits. Finally, we assess the influence of astrocytes on network performance. The results show that astrocytes significantly improve network performance and facilitate faster learning compared to conventional spiking neural networks (SNNs) alone.
<figure><img src="image_4.png" /></figure>

Figure 3: The tripartite synapse model consists of two layers of neurons ($n_{\text{number}}^1$) and an astrocytic layer ($a_{\text{number}}^{\text{layer}}$) in between. The presynaptic neuron ($n^1$) is a Poisson spiking neuron whose spike-based event modulates the postsynaptic neurons ($n^2$) and the astrocytes ($a^1$) via synapses ($s^1$). The postsynaptic layer consists of integrate-and-fire (IF) neurons with lateral inhibitions from interneurons (IN, $m^2$). Therefore, the postsynaptic neuron receives three types of inputs whose strengths are defined by the synaptic weights $\omega_{\text{exc1}}$, $\omega_{\text{exc2}}$, and $\omega_{\text{inh}}$ from the presynaptic neuron through neurotransmission, the astrocyte through gliotransmission, and inhibition from neighboring IN. The IN sense the spiking activity of the coupled postsynaptic neuron and relays it to the neighboring excitatory neurons. The astrocyte then integrates the synaptic inputs whose coupling strengths are controlled by $\beta$. Here, the model ensures that in a tripartite connection, the astrocyte modulates the postsynaptic neuron counterpart of its input neuron.

## 2 SNAN Architecture

This section describes the tripartite synaptic connection, the neuron and astrocyte layers, and the corresponding network architecture. It also includes the learning algorithm used to train the network, the simulation process, and the performance analysis method.

### 2.1 Tripartite Synapse Model.
In the proposed network architecture, as depicted in Figure 3, the tripartite synapses comprise Poisson spiking presynaptic neurons ($n^1$) that are coupled to the astrocyte ($a^1$) and the postsynaptic neuron ($n^2$) via the synapses ($s^1$).

#### 2.1.1 Neuron Models.
The spiking rate of the Poisson neurons depends on the input image pixel intensity, which ranges from 0 for black to 31.875 Hz
for white pixels. The rate is chosen to ensure that the presynaptic inputs do not oversaturate the astrocytic dynamics while generating a postsynaptic spike.

The postsynaptic neuron integrates three types of inputs: (1) fast excitatory inputs arising from neurotransmission, (2) slow excitatory inputs due to gliotransmission, and (3) inhibitory inputs originating from the interneurons ($m^2$). The postsynaptic neuron potential ($v$) in mV is governed by the adaptive integrate-and-fire (AIF) neuron model (Brette & Gerstner, 2005) given as follows:

$$ 
\begin{aligned}
\tau \frac{dv}{dt} &= (v_{\text{rest}} - v) + g_{\text{exc1}} (v_{\text{exc1}} - v) \\
&\quad + g_{\text{exc2}} (v_{\text{exc2}} - v) + g_{\text{inh}} (v_{\text{inh}} - v), \\
\tau_x \frac{dx}{dt} &= -x, \quad x = g_{\text{exc1}}, g_{\text{exc2}}, g_{\text{inh}}.
\end{aligned}
\qquad (2.1) $$

The conductance (in nS) of the fast excitatory, slow excitatory, and inhibitory synapses are equal to the weight summations $g_{exc1}$, $g_{exc2}$, and $g_{inh}$, respectively, and where $x$ defines their conductance decay through time $t$. The parameter $v_{rest}$ is the membrane resting potential while $v_{exc1}$, $v_{exc2}$, and $v_{inh}$ are the reversal potentials of the excitatory and inhibitory synapses. The constant parameters, $\tau$ and $\tau_x$, are the time constants of the membrane potential and the synapses, respectively. When the AIF neuron membrane potential $v$ crosses its spiking threshold $\theta$, it generates a spike, then immediately resets to a level equal to $v_{reset}$ and remains here until the refractory period, $\tau_{ref}$, ends. During training, the adaptive threshold increases by $\alpha'$ after a spike and decays following the exponential model

$$ \frac{d\theta}{dt} = \frac{\theta - \theta_0}{\tau_{\theta}}, \qquad (2.2) $$

where $\theta = v - v_{reset}$ and $\tau_{\theta}$ is the time constant.

The postsynaptic neuron layer consists of lateral inhibition for maintaining homeostasis and signaling the postsynaptic neuron activity to its neighbors. Here, the postsynaptic neuron signals its spiking activity to the IN, and in return, the IN then inhibits the activity of the other neurons, generating competition among neurons. The interneurons follow the IF model in equation 2.1 but without the adaptive property $\theta$ and the $g_{exc2}$ and $g_{inh}$ conductances, while $g_{exc1}$ changes with respect to the postsynaptic neuron spiking activity. Table 1 lists the parameters of the postsynaptic excitatory and inhibitory neurons.

### 2.1.2 Astrocyte Model.
The astrocyte functions as a point process that integrates presynaptic inputs, following the simplified model proposed by
<table><caption>Table 1: Parameters of the Postsynaptic Neurons with Lateral Inhibition.</caption><thead><tr><th>Parameter</th><th>Value</th><th>Description</th></tr></thead><tbody><tr><td colspan="3"><strong>Postsynaptic excitatory neurons</strong></td></tr><tr><td>v<sub>rest</sub></td><td>-65 mV</td><td>Membrane resting potential</td></tr><tr><td>v<sub>exc1</sub></td><td>0 mV</td><td>Fast synaptic input reversal potential</td></tr><tr><td>v<sub>exc2</sub></td><td>0 mV</td><td>Slow synaptic input reversal potential (Tewari & Parpura, 2013)</td></tr><tr><td>v<sub>inh</sub></td><td>-100 mV</td><td>Inhibitory synaptic reversal potential</td></tr><tr><td>τ</td><td>100 ms</td><td>Membrane time constant</td></tr><tr><td>τ<sub>exc1</sub></td><td>1 ms</td><td>Excitatory synapse (fast) time constant</td></tr><tr><td>τ<sub>exc2</sub></td><td>600 ms</td><td>Excitatory synapse (slow) time constant (Tewari & Parpura, 2013)</td></tr><tr><td>τ<sub>inh</sub></td><td>2 ms</td><td>Inhibitory synapse time constant</td></tr><tr><td>v<sub>reset</sub></td><td>-65 mV</td><td>Reset potential</td></tr><tr><td>v<sub>thresh</sub></td><td>-52 mV</td><td>Spiking threshold level</td></tr><tr><td>t<sub>ref</sub></td><td>5 ms</td><td>Refractory period</td></tr><tr><td>θ<sub>0</sub></td><td>20 mV</td><td>Spiking threshold offset</td></tr><tr><td>τ<sub>θ</sub></td><td>107 ms</td><td>Spiking threshold time constant</td></tr><tr><td>α'</td><td>0.05 mV</td><td>Increase in spiking threshold</td></tr><tr><td colspan="3"><strong>Postsynaptic inhibitory neurons</strong></td></tr><tr><td>v<sub>rest</sub></td><td>-60 mV</td><td>Membrane resting potential</td></tr><tr><td>v<sub>exc1</sub></td><td>0 mV</td><td>Synapse reversal potential</td></tr><tr><td>τ</td><td>100 ms</td><td>Membrane time constant</td></tr><tr><td>τ<sub>exc1</sub></td><td>1 ms</td><td>Excitatory synapse time constant</td></tr><tr><td>v<sub>reset</sub></td><td>-45 mV</td><td>Reset potential</td></tr><tr><td>v<sub>thresh</sub></td><td>-40 mV</td><td>Spiking threshold</td></tr><tr><td>t<sub>ref</sub></td><td>2 ms</td><td>Refractory period</td></tr></tbody></table>

Note: Unless otherwise stated, the parameters are taken from Diehl and Cook (2015).

Postnov et al. (2009, 2011). We chose the Postnov model for several key reasons: (1) It captures both fast Ca²⁺ oscillations and slower dynamics in astrocytes (Postnov et al., 2009). (2) It models astrocytic activity over multiple temporal scales (Postnov et al., 2011). (3) The model effectively couples astrocytic activity with neuronal dynamics (Amiri et al., 2011). (4) Importantly, it provides a balance between biological accuracy and computational simplicity. While more complex models, such as the Li-Rinzel model (Li & Rinzel, 1994) and its adaptations (Manninen et al., 2018), provide greater biological detail, they are computationally expensive. In contrast, simpler models, such as those based on the Hill function (Cresswell-Clay et al., 2018), may reduce simulation complexity but are limited in biological accuracy and temporal and spatial resolution. Therefore, due to the properties of the Postnov model, it is well suited for large-scale networks and real-time tasks such as image recognition.

During neurotransmission, when the presynaptic neuron releases Glu⁻ into the synaptic cleft, a fraction of the [Glu⁻] spills over into the extrasynaptic space, activating astrocytic mGluRs, as illustrated in Figure 1.
Here, the synaptic coupling variable z reflects the activation of astrocytic mGlur5; it spikes to 1 and then quickly inactivates, decaying back to 0 in synchrony with the presynaptic neuron firing. The coupling variable z (from synapse to astrocyte) follows the simplified model,

$$ \tau_S \frac{dz}{dt} = -\frac{z}{d_S}, \qquad (2.3) $$

where $\tau_S$ represents the synaptic delay. The parameter $d_S$ controls the relaxation time of z. It represents the decrease in the astrocytic response with the synaptic activity and can be influenced by factors that determine the astrocyte activation time such as receptor desensitization and intracellular signaling pathways.

The activation of astrocytic mGluRs by extrasynaptic Glu⁻ (z) triggers the production of IP₃, whose concentration is represented by $S_m$. The dynamics of $S_m$ are governed by the following nonlinear equation:

$$ \tau_{S_m} \frac{dS_m}{dt} = (1 + \tanh(s_{S_m}(z - h_{S_m}))) (1 - S_m) - \frac{S_m}{d_{S_m}}, \qquad (2.4) $$

where the parameter $\tau_{S_m}$ controls the timescale of IP₃. Equation 2.4 models the nonlinear, threshold-dependent nature of IP₃ production. Biologically, this means that IP₃ production does not start immediately at low glutamate levels; instead, there is a threshold level of Glu⁻ ($h_{S_m}$) needed to trigger significant increase in IP₃ production. Once the IP₃ concentration ($S_m$) reaches this threshold, IP₃ production increases rapidly as defined by the steepness parameters of the activation curve ($S_{S_m}$). Then $d_{S_m}$ governs the deactivation rate of IP₃, determining how quickly the astrocyte reduces or clears IP₃ concentration when extrasynaptic Glu⁻ input decreases. This model captures the dynamic production and degradation of IP₃ in response to fluctuating Glu⁻ levels in the extracellular space.

Let us consider an astrocytic compartment in a tripartite synapse (see Figure 1). An astrocyte exhibits fast Ca²⁺ oscillation correlated with the production and degradation of IP₃ molecules. The astrocyte also has calcium stores called endoplasmic reticulum (ER) that releases and reuptakes Ca²⁺ molecules to and from the intracellular space. This Ca²⁺ exchange results in a slow Ca²⁺ dynamics. Therefore, the total Ca²⁺ concentration in the astrocytic compartment is the summation of the current intracellular [Ca²⁺] (c), the Ca²⁺ increase due the IP₃ production, and the Ca²⁺ exchange between the intracellular space and the ER. The intracellular Ca²⁺ dynamics in the astrocyte are described by the following system,

$$ \tau_c \frac{dc}{dt} = c - c_4 f(c, c_e) + (r + \beta S_m), $$

$$ \epsilon_c \tau_c \frac{dc_e}{dt} = f(c, c_e), $$
<table><caption>Table 2: Simplified Astrocytic Model Parameters.</caption><thead><tr><th>Parameter</th><th>Value</th><th>Description</th></tr></thead><tbody><tr><td>Neuron-astrocyte coupling</td><td></td><td></td></tr><tr><td>τ<sub>S</sub></td><td>10 ms</td><td>Synaptic delay time constant</td></tr><tr><td>s<sub>S</sub></td><td>1</td><td>Steepness of activation</td></tr><tr><td>h<sub>S</sub></td><td>0</td><td>Activation control</td></tr><tr><td>d<sub>S</sub></td><td>3</td><td>Relaxation control</td></tr><tr><td>IP<sub>3</sub> production</td><td></td><td></td></tr><tr><td>τ<sub>Sm</sub></td><td>100 ms</td><td>Time constant</td></tr><tr><td>s<sub>Sm</sub></td><td>100</td><td>Steepness of activation</td></tr><tr><td>h<sub>Sm</sub></td><td>0.45</td><td>Threshold</td></tr><tr><td>d<sub>Sm</sub></td><td>3</td><td>Deactivation rate</td></tr><tr><td>Astrocytic Ca<sup>2+</sup></td><td></td><td></td></tr><tr><td>ε<sub>c</sub></td><td>0.04</td><td>Characteristic time control</td></tr><tr><td>c<sub>1</sub></td><td>0.13</td><td>Ca<sup>2+</sup> oscillation control parameter</td></tr><tr><td>c<sub>2</sub></td><td>0.9</td><td>Ca<sup>2+</sup> oscillation control parameter</td></tr><tr><td>c<sub>3</sub></td><td>0.004</td><td>Ca<sup>2+</sup> oscillation control parameter</td></tr><tr><td>c<sub>4</sub></td><td>2/ε<sub>c</sub></td><td>Ca<sup>2+</sup> oscillation control parameter</td></tr><tr><td>r</td><td>0.31</td><td>Ca<sup>2+</sup> oscillation initial state</td></tr><tr><td>τ<sub>c</sub></td><td>600 ms</td><td>Time constant for Ca<sup>2+</sup> oscillation</td></tr><tr><td>c<sub>thresh</sub></td><td>0.4</td><td>Threshold of astrocytic activation</td></tr></tbody></table>

Note: The parameters are taken from Postnov et al. (2009, 2011).

$$f(c, c_e) = c_1 \frac{c^2}{1+c^2} - \left( \frac{c_e^2}{1+c_e^2} \right) \left( \frac{c^4}{c_2^4+c^4} \right) - c_3 c_e, \quad (2.5)$$

where c represents the current [Ca²⁺] at time t. The second term, c₄f(c, c_e), represents the Ca²⁺ flux between the ER and the intracellular space, where f(c, c_e) models the complex dynamics of Ca²⁺ release and reuptake of the ER, whose concentration is represented by c_e. The third term, r + βSₘ, captures the influence of IP₃ on Ca²⁺ elevation, where r reflects the initial Ca²⁺ oscillation state and β scales the effect of IP₃ on the [Ca²⁺]. Refer to Table 2 for details on the constant parameters.

The astrocyte enwraps multiple synapses through its astrocytic processes, as shown in Figure 2. Intracellular Ca²⁺ and IP₃ molecules from one astrocytic process diffuse to the neighboring compartments or to the astrocytic soma. This molecular diffusion allows the astrocyte to integrate the presynaptic signals from multiple synaptic connections, where β is the control parameter defining the magnitude of influence of the presynaptic neuron activity on the astrocytic Ca²⁺. The modified Ca²⁺ model (Postnov et al., 2011) is defined as

$$\tau_c \frac{dc}{dt} = c - c_4 f(c, c_e) + \left( r + \sum_{i=1}^{k} \beta S_{m_i} \right). \quad (2.6)$$
<figure><img src="image_5.png" /><figcaption>Figure 4: Network architecture for image recognition is a three-stage process comprising an input preprocessing unit, a SNAN learning unit, and a classifier unit.</figcaption></figure>

When the astrocytic [Ca²⁺], c, increases beyond a certain threshold, c<sub>thresh</sub>, it triggers the release of extrasynaptic Glu⁻, activating slow excitatory synapses of the coupled postsynaptic neurons. After the Glu⁻ spike, the astrocyte stays in the refractory period provided that c stays above the threshold. The astrocytic parameters are listed in Table 2.

In the tripartite synapse model (see Figure 1), astrocytes serve as a third active component in neural processing, along with pre- and postsynaptic neurons. Their morphological complexity and Ca²⁺-dependent excitability give them a critical role in signaling. While astrocytic signaling is slower than neuronal signaling—operating on a scale of seconds to minutes (Haydon & Nedergaard, 2015)—astrocytes share computational features with neurons. Neurons transmit information via fast action potentials, whereas astrocytes use Ca²⁺-based chemical signaling (see equation 2.5), releasing gliotransmitters into the extrasynaptic space. But unlike neurons, which integrate signals through summation of electrochemical potentials, astrocytes rely on Ca²⁺ diffusion across processes to integrate synaptic inputs (Figure 2 and equation 2.6; see also Cresswell-Clay et al., 2018). Astrocytes modulate synaptic transmission by influencing pre- and postsynaptic activity, affecting both short- and long-term plasticity and wider network dynamics. They also connect with thousands of synapses, bridging communication between unconnected neurons. This broad connectivity, along with their ability to laterally regulate synaptic signals, positions astrocytes as computational units that expand the capacity and complexity of neural networks.

**2.2 Network Architecture.** The digit recognition process follows the network architecture presented in Figure 4. The process starts from the image preprocessing unit, where the input or sample image is converted
into rate-coded signals, followed by unsupervised learning by the SNAN and based on the SNAN output spike patterns, the machine learning (ML) model predicts the output digit. Given a sample digit from the MNIST data set a 28×28-pixel image of a handwritten digit is converted into a 784 × 1 vector of pixel intensities. One pixel intensity corresponds to the spiking frequency that drives a Poisson neuron in the input layer of the SNAN.

In the spiking network, 784 input layer neurons form 78,400 fully con-
nected (dense) synapses with the neurons in the first layer. There is a 1:1
ratio between the number of excitatory and inhibitory neurons in the first
layer, where the forward connection (from the excitatory to the inhibitory)
is a one-to-one topology. At the same time, the lateral inhibitions form 9900
inhibitory synapses, where an IN connects with all excitatory neurons in
the same layer except with the one in the forward connection. The astro-
cytic layer between the two neuronal layers, also following a 1:1 ratio with
the first layer neurons, forms the tripartite synapses by connecting with the
synapses rather than directly with the input layer neurons. Because an as-
trocyte is coupled with the synapses, it can receive multiple presynaptic in-
puts from the same input layer neuron and increase its influence on a first
layer neuron by modulating multiple tripartite synapses coupled to that
postsynaptic neuron.

In the classifier unit, the spike counter converts the output spiking pat-
terns of the first layer excitatory neurons into a vector of whole numbers
corresponding to the number of spikes in each neuron. This vector gives the
set of input features for the classifier. Our simulation analysis suggests that
the cosine k-NN classifier using five-fold cross-validation yields a faster and
better classification performance than the other available machine learning
classifier. Finally, the network predicts the output from 10 classes labeled
from 0 to 9. This paradigm requires that the SNAN generates a stable spik-
ing pattern for each input class for the network to recognize and differenti-
ate each input pattern.

**2.3 Spike-Timing-Dependent Plasticity.** A form of long-term plasticity called spike-timing-dependent plasticity (STDP) addressed the temporal issue in Hebbian plasticity (Caporale & Dan, 2008; Frémaux & Gerstner, 2016; Markram et al., 2012). Rather than correlated inputs driving the synaptic plasticity, the spiking activities of neurons within a temporal window define the direction of synaptic plasticity, and the relative timing between the pre- and postsynaptic firings determines the change in the synaptic weights. Therefore, the STDP learning rule is suitable for training the SNAN due to the significant difference between the neuronal and astrocytic time scales.

In the STDP rule following Diehl and Cook's (2015) model, the increase
in synaptic weight includes a weight-dependent term so that during postsy-
naptic activation, the weight increases only after the presynaptic activation.
Let the traces αpre, αpost1, and αpost2 decay exponentially by
$$
\begin{align}
\tau_{pre} \frac{d\alpha_{pre}}{dt} &= -\alpha_{pre}, \nonumber \\
\tau_{post1} \frac{d\alpha_{post1}}{dt} &= -\alpha_{post1}, \nonumber \\
\tau_{post2} \frac{d\alpha_{post2}}{dt} &= -\alpha_{post2}, \tag{2.7}
\end{align}
$$

where $\tau_{pre}$, $\tau_{post1}$, and $\tau_{post2}$ are the decay constants equal to 20 ms, 20 ms, and 40 ms, respectively. On presynaptic activation, the weights are updated following the rule

$$w \rightarrow w + \eta_{pre} \alpha_{post1}, \tag{2.8}$$

while on the postsynaptic activation,

$$w \rightarrow w + \eta_{post} \alpha_{pre} \alpha_{post2}, \tag{2.9}$$

where $\eta_{pre}$ and $\eta_{post}$ are the pre- and postsynaptic rates, respectively. The excitatory-to-excitatory connections use the pre- and postsynaptic spiking weight update rules ($\omega_{exc1}$ increases when the presynaptic voltage is above $v_{thresh}$), while astrocyte-postsynaptic coupling increases postsynaptic weights only during astrocytic activation ($\omega_{exc2}$ is updated when $c$ increases beyond $c_{thresh}$). Due to the significant timescale difference between neurons and astrocytes, the slow synapse conductance depletes to zero when the postsynaptic neuron fires continuously. Moreover, the excitatory-to-inhibitory and the inhibitory-to-excitatory $\Delta w$ are constants equal to 17 and 10.4, respectively.

**2.4 Simulation Method and Performance Analysis.** Here, we used the Brian2 simulator (Goodman & Brette, 2009), in conjunction with the Python programming language, as it explicitly describes models in a high-level form by writing the differential equations directly in the code and allows us to efficiently model and integrate astrocytes into the network.

The MNIST data set (Deng, 2012) contains 60,000 training, 5000 validation, and 5000 test images. Each image is fed into the proposed network one at a time for 350 ms, followed by a resting time of 150 ms, to ensure that the network activity due to the previous image will not overlap with the new sample; therefore, one sample takes 500 ms to pass through the SNAN. The network continuously updates its weights via the STDP learning rule and the neuronal firing thresholds during training. We divide one epoch (containing all the training images) into batches of 1000 samples each and then normalize the weights after each batch. After each epoch, the validation set is fed into the network using the learned parameters.
<table><caption>Table 3: Set of Hyperparameters per Configuration.</caption><thead><tr><th>Hyperparameters</th><th>SNN</th><th>SNAN1</th><th>SNAN2</th><th>SNAN3</th></tr></thead><tbody><tr><td>η<sub>exc1 pre</sub></td><td>0.0001</td><td>0.0001</td><td>0.0001</td><td>0.0001</td></tr><tr><td>η<sub>exc1 post</sub></td><td>0.01</td><td>0.01</td><td>0.01</td><td>0.01</td></tr><tr><td>η<sub>exc2 pre</sub></td><td>—</td><td>0.000001</td><td>0.00001</td><td>0.000001</td></tr><tr><td>η<sub>exc2 post</sub></td><td>—</td><td>—</td><td>—</td><td>—</td></tr><tr><td>w<sub>exc1 max</sub></td><td>1</td><td>1</td><td>1</td><td>1</td></tr><tr><td>w<sub>exc2 max</sub></td><td>—</td><td>0.001</td><td>0.01</td><td>0.001</td></tr><tr><td>β</td><td>—</td><td>0.00005</td><td>0.00005</td><td>0.0001</td></tr></tbody></table>

We simulated different spiking network configurations to determine the hyperparameters that lead to optimum network performance: one SNN and three SNANs (SNAN,1, SNAN2, SNAN3). All networks have the same number of neurons and astrocytes (except in SNN), while the number of synaptic connections with astrocytic coupling increases from 10% to 60% (increment of 10%) of the total synapses, where each astrocyte receives the same number of presynaptic inputs. The set of hyperparameters is listed in Table 3. The simulations were performed using the high-performance computing resources from the Barcelona Computing Center and the University of Burgundy Computing Center.

**2.5 Performance Analysis.** The classifier accuracy defines how well the system groups and distinguishes the spiking patterns generated by the SNAN from one class to another. Therefore, the classification performance depends on the extent to which the SNAN learned the rate-coded images. During training, we determine the accuracy after each epoch. Then, using the learned parameters after each epoch, we feed the validation set into the network for prediction and compare the results with the training accuracy. Therefore, the parameters chosen for the test network simulations are the learned parameters of the epoch giving the maximum average accuracy between the train and validation sets.

## 3 Results

We simulated the spiking networks and set the SNN as the baseline of the SNAN activities. Here, we focus on the influence of astrocytes in network activities. The minimum numbers of neurons for conventional MNIST classification networks are 784 and 100 in the input and first layers, respectively. Here, the number of astrocytes in all networks is equal to the number of first layer excitatory neurons (1:1 ratio); however, the number of synapses covered by astrocytes increases by 10% of the total number of input to first layer synaptic connections. A higher number of neurons results in more synapses.
<figure><img src="image_6.png" /><figcaption>Figure 5: Sample tripartite synapse dynamics. Parameters: $dt = 1$ ms, input spike rate = 5 Hz, $v_{rest} = -70$ mV, $\theta = -40$ mV, $c_{thresh} = 0.40$, $g_{e1} = 1$ nS, synaptic delay = 9.62 ms, $\beta = 0.006$, $\tau_c = 50$ ms, $g_{e2} = 1$ nS.</figcaption></figure>

Moreover, astrocytes create two more types of synaptic coupling (presynaptic neuron-to-astrocyte and astrocyte-to-postsynaptic neuron shown in Figure 3), resulting in a more complex network that requires prolonged simulations.

The SNAN design is a highly iterative process, requiring numerous trials before arriving at the possible set of hyperparameters shown in Table 3. The neuronal and neuron-to-neuron synaptic hyperparameters are consistent in all the networks. Based on our initial simulations, these parameters lead to regular neuronal spiking. The Poisson neuron changes its spiking rate every 500 ms. We also determined that the Poisson neuronal firing rate lies between 0 to 32 Hz (proportional to the pixel intensity from 0 to 255) to prevent oversaturation of astrocytic $S_m$ (IP₃ pathway) and avoid overexcitation of astrocytic Ca²⁺ dynamics, c.

**3.1 Tripartite Synapse Dynamics.** Figure 5 is an overview of the activities of a single tripartite synapse without STDP and wherein the astrocyte only receives a presynaptic input and modulates one postsynaptic neuron.
The presynaptic neuron generates 5 Hz and regularly spaced spikes. The synaptic coupling variable *z* increases to one simultaneously with the input neuron spikes and then exponentially decays back to zero, replicating the neurotransmitter release and recycling process. In addition, the IP₃ pathway (*S*<sub>m</sub>) also increases and decays with *z*, and whose peak amplitudes stabilize at 0.54. The Ca²⁺ variable *c* generates spikes, on crossing the 0.4 threshold, with a peak amplitude equal to 1.29 with a pulse duration of 58 ms. After activation, the astrocyte is in the refractory period provided that *c* > 0.4. A single fast excitatory synaptic input generates a postsynaptic spike, *v*, with -64.38 mV peak amplitude (first three spikes). In this example, the fast and slow excitatory synapse conductances and weights are equal to 1 nS and 1, respectively, creating the same strength of influence on the postsynaptic neuron. However, the combination of the neuronal and astrocytic inputs triggers a sudden increase in *v* up to -46.35 mV. Then *v* decays to 0 while fluctuating, resulting from the fast activation of presynaptic input.

In the SNAN, astrocytes receive hundreds of inputs; therefore, the sum-
mation of slow synaptic weights must generate a conductance, gexc2, low
enough to not overexcite the postsynaptic neuron. In addition, the neuron-
astrocyte coupling control factor β must be high enough to trigger astro-
cytic activation. We performed multiple simulations and determined the
astrocytic hyperparameters in Table 3. Here, the astrocyte-neuron coupling,
*w*exc2, is updated upon the arrival of astrocytic input; however, there is
no *w*exc2 update (ηexc2post is null) upon postsynaptic neuron spiking due to
the significant timescale difference between the astrocyte and the neuron.
If STDP is bidirectional, then the fast spiking of the postsynaptic neuron
can cancel the astrocyte-neuron coupling. High-slow excitatory synaptic
weights excite the first layer neurons that can cause erratic spiking patterns,
suggested by the average spiking rates of SNAN2 in Figure 6a that grow
with an extended number of tripartite connections. The astrocytic popula-
tion rates of SNAN1 and SNAN2 (see Figure 6b) have the same trend due to
the similarity in β. β = 0.001 and β = 0.0005, with 10% and 20% tripartite
synapses, respectively, are inadequate to activate the astrocytes.

**3.2 Training Results.** We trained the networks by presenting the entire 60,000 MNIST training images for a duration spanning 25 epochs. The synaptic weight evolution in Figure 7 shows the ability of the first layer neurons to learn the representative inputs. The patterns suggest which input neurons were inactive and strongly coupled with the first layer neurons. Here, the 784 excitatory-to-excitatory synaptic weights vector of a neuron is rearranged to a 28×28 matrix. For example, the second neuron in the SNN learned the features of class 1, and the ninth neuron in SNAN3 initially learned class 8 features, then became strongly coupled with class 1 inputs. The receptive fields show that neurons in the three SNAN configurations learn almost the same features, while the neurons in SNN learn differently.
<figure><img src="image_7.png" /><figcaption>Figure 6: Population rates of (a) first layer neurons and (b) astrocytic layers with an increasing number of tripartite synaptic connections.</figcaption></figure>

<figure><img src="image_8.png" /><figcaption>Figure 7: Receptive field evolutions for the nine first layer neurons of the SNN and SNANs with 50% tripartite synaptic components.</figcaption></figure>

The classifier then predicts the output class based on the spiking patterns given by the SNAN per individual input. Figure 8 summarizes the change in training accuracy of each network configuration with increasing synapses covered by astrocytes. The curve with 0% tripartite synapses is the training accuracy of SNN, used as the reference in comparing the SNAN performance. The accuracy curves are the same during initial training, and the SNN gradually outperforms the SNANs. In SNAN2, it is noticeable that the rate of increase in the accuracy of SNN is maximum. However, starting from epoch 13, the SNN performance decreases to a minimum and becomes
<figure><img src="image_9.png" /><figcaption>Figure 8: Training accuracy per network configuration (a) SNAN1, (b) SNAN2, and (c) SNAN3 with an increasing number of tripartite synapses as a function of the presented training epoch.</figcaption></figure>
stable at an average accuracy of 65.38%, and then all SNANs achieve higher accuracy than SNN. Figure 7 also indicates that SNANs with 50% tripartite synapses have higher performance in all network configurations, wherein the maximum accuracy is 69.94% at epoch 13 for SNAN1, 69.76% at epoch 25 for SNAN2, and 70.13% at epoch 23 for SNAN3. From these results, SNAN3 with 50% tripartite synapses leads to maximum classification performance. It is also indicative that the SNAN produces spiking patterns easily categorized by the classifier unit.

The validation accuracy curves indicate the reverse: that the SNN best predicts the output class with maximum accuracy of 70.66% at epoch 25. In SNAN1 (10%–50% tripartite synapses), the accuracy curves stabilize at an average level equal to 61.60%, almost 9% lower than SNN. In SNAN2, where the astrocyte-neuron synaptic coupling strength is a factor of 10 greater than SNAN1, the increase of tripartite connections to 50% and 60% results in a sudden decrease in validation accuracy, implying overfitting of the trained SNAN2 and that the introduction of unknown samples generates irregular spiking patterns. SNAN3 has the same coupling strength as SNAN1 but double the presynaptic-to-astrocytic coupling strength. In this case, the validation accuracy increases gradually but results in higher performance (for 20% and 30% tripartite synapses).

We chose the test network parameters from the train versus validation accuracy graphs in Figures 9 to 11. Strikingly, the SNN network has a validation accuracy curve of 6% higher than the training accuracy, indicating underfitting. The increase of astrocytic coverage by up to 30% diminishes the difference between the train and validation accuracy of SNAN1 (see Figure 9). A further increase to 60% results in considerable divergence between the accuracy curves. The same results are noticeable in SNAN2 (see Figure 10); validation accuracy increases with more astrocytic-mediated synapses but drops significantly to 10% accuracy for 60% tripartite connections. However, SNAN3 validation accuracy, with slow astrocyte-to-neuron STDP and stronger neuron-to-astrocyte coupling ($\beta = 0.001$), also shows a slower convergence with the training accuracy (see Figure 11). The SNAN3 with 20% and 30% tripartite connections have comparable training and validation accuracy levels in all configurations. Also, for the given SNAN3 parameters with 40% to 60% astrocytic mediated synapses, the network performance improves gradually, compared to the low validation accuracy in SNAN2 (see Figure 10). These results are in accord with the results of the biologically plausible model, suggesting that astrocytes can either improve or impair network activities (Lorenzo et al., 2020, 2022; Vuillaume et al., 2021).

The network parameters giving the optimal model complexity (balance between variance and bias) are the learned parameters after the epochs specified in Table 4. The SNN is optimal after 10 times input presentations. The SNN, in this case, displays underfitting (having a higher validation than training accuracy). SNANs achieve higher network performance and, in some instances, require less training. For example, it takes only seven
<figure><img src="image_10.png" /><figcaption>Figure 9: SNAN1 training (solid lines) and validation (dashed lines) accuracy for n% tripartite synapses. Parameters: $\eta_{exc1pre} = 0.0001$, $\eta_{exc1post} = 0.01$, $\eta_{exc2pre} = 0.000001$, $w_{exc1max} = 1$, $w_{exc2max} = 0.001$, and $\beta = 0.00005$.</figcaption></figure>

<figure><img src="image_11.png" /><figcaption>Figure 10: SNAN2 training (solid lines) and validation (dashed lines) accuracy for n% tripartite synapses. Parameters: $\eta_{exc1pre} = 0.0001$, $\eta_{exc1post} = 0.01$, $\eta_{exc2pre} = 0.00001$, $w_{exc1max} = 1$, $w_{exc2max} = 0.01$, and $\beta = 0.0005$.</figcaption></figure>
<figure><img src="image_12.png" /><figcaption>Figure 11: SNAN3 training (solid lines) and validation (dashed lines) accuracy for n% tripartite synapses. Parameters: $\eta_{exc1pre} = 0.0001$, $\eta_{exc1post} = 0.01$, $\eta_{exc2pre} = 0.000001$, $w_{exc1max} = 1$, $w_{exc2max} = 0.001$, and $\beta = 0.001$.</figcaption></figure>

<table><caption>Table 4: Epochs of the Network Parameters with Optimal Model Complexity.</caption><thead><tr><th rowspan="2">% Tripartite Synapse</th><th colspan="2">SNAN1</th><th colspan="2">SNAN2</th><th colspan="2">SNAN3</th></tr><tr><th>Train Accuracy (%)</th><th>Epoch #</th><th>Train Accuracy (%)</th><th>Epoch #</th><th>Train Accuracy (%)</th><th>Epoch #</th></tr></thead><tbody><tr><td>0</td><td>67.76</td><td>10</td><td>67.76</td><td>10</td><td>67.76</td><td>10</td></tr><tr><td>10</td><td>67.30</td><td>14</td><td>67.30</td><td>14</td><td>67.94</td><td>25</td></tr><tr><td>20</td><td>67.12</td><td>19</td><td>65.90</td><td>12</td><td>66.65</td><td>20</td></tr><tr><td>30</td><td>68.94</td><td>18</td><td>66.34</td><td>12</td><td>68.41</td><td>19</td></tr><tr><td>40</td><td>67.56</td><td>7</td><td>67.45</td><td>24</td><td>67.45</td><td>25</td></tr><tr><td>50</td><td>69.26</td><td>15</td><td>68.83</td><td>10</td><td>66.36</td><td>25</td></tr><tr><td>60</td><td>67.13</td><td>25</td><td>67.07</td><td>24</td><td>68.97</td><td>9</td></tr></tbody></table>

iterations for SNAN1 with 40% tripartite connections to achieve an accuracy of 0.20% less than SNN and nine iterations for SNAN3 with 60% tripartite connections to obtain 1.21% higher than SNN. In addition, for the same number of epochs, SNAN2 with 50% tripartite connections achieves 1.07% higher than SNN. The results in Table 4 suggest that astrocytes support neuronal memory formation and promote faster learning.

**3.3 Test Results.** We tested the network using the 5000 test images from the MNIST data set. The test simulations take an average of 0.41 s, 0.69 s, 0.88 s, 1.02 s, 1.33 s, 1.67 s, and 1.59 s for the networks with 0%, 10%, 20%,
<table><caption>Table 5: Test Accuracy.</caption><thead><tr><th>% Tripartite Synapse</th><th>SNAN1</th><th>SNAN2</th><th>SNAN3</th></tr></thead><tbody><tr><td>0</td><td>82.46</td><td>82.46</td><td>82.46</td></tr><tr><td>10</td><td>65.46</td><td>65.46</td><td>66.80</td></tr><tr><td>20</td><td>67.54</td><td>66.44</td><td>71.96</td></tr><tr><td>30</td><td>68.94</td><td>69.98</td><td>75.28</td></tr><tr><td>40</td><td>67.26</td><td>67.32</td><td>70.26</td></tr><tr><td>50</td><td>68.56</td><td>21.66</td><td>72.44</td></tr><tr><td>60</td><td>71.20</td><td>11.28</td><td>71.96</td></tr></tbody></table>

30%, 40%, 50%, and 60% tripartite synaptic connections, respectively, to predict the class of a single test image. Table 5 shows the resulting test accuracy per network configuration. The proposed SNN gives an accuracy of 82.46%, 0.44% less than the accuracy reported by Diehl and Cook (2015) for their network with 100 first layer neurons. Of the neuron-astrocyte networks, the parameters of SNAN3 give the best network performance, with maximum accuracy of 75.28% for 30% tripartite connections. Indeed, the neural network displays higher prediction accuracy than the neuron-astrocyte networks. However, the neuron network also displays high bias and underfitting, given that the test accuracy is 14.17% greater than the training accuracy. It suggests the SNN network is not yet fully optimized and requires extended training for more than 25 epochs. Therefore, the astrocytes improve network performance, especially in the SNAN3, where there is an optimum variance-bias trade-off.

The diagonals of the confusion matrices in Figure 12 show the number of the correctly predicted input class. Even though the SNN correctly predicted class 0 more times than the SNAN, the SNAN still generates higher precision (92%) for the specified class, given that there are more instances that the SNN confuses the remaining input classes as 0. The SNAN precisely predicts input classes 1 (93.70%) and 3 (97.10%). Furthermore, The SNAN also displays higher recall for input classes 4 (79.90%) and 6 (95.60%), correctly predicting the input class and not confusing them with other classes. For other instances, the SNN displays higher precision and recall compared to the SNAN as expected. SNAN3 also predicts input classes 1 and 2 as 8. This prediction can be caused by the receptive field of the neuron showing combined features of 1 and 8; for example, the ninth neuron in the SNAN3 shown in Figure 7. SNAN3 also confuses class 9 as 4 due to their similar vertical features. The neurons with receptive fields of 9 spike at a high rate when the number 4 is presented to the network.

## 4 Discussion

We have proposed a novel and simplified spiking neuron-astrocyte network for unsupervised learning using STDP, combined with k-NN
<figure><img src="image_13.png" /><figcaption>Figure 12: Confusion matrices for (a) SNN and (b) SNAN3 with 30% tripartite synapses. The precision and recall are displayed, respectively, below and on the right of the confusion matrix.</figcaption></figure>

classifier for image recognition. The input layer consists of Poisson spiking neurons translating the input image pixel intensity into spike events. The dense synaptic connections transmit the input activities to the 100 post-synaptic neurons. The postsynaptic neurons then learn the input features by changing their synaptic weights following the STDP algorithm. In these networks, astrocytes have three main functions: (1) create a new network layer for integrating and translating presynaptic inputs, (2) act as additional synaptic connections simplifying network architecture, and (3) modulate synaptic plasticity for faster learning. We have achieved a maximum of 75.28% of classification performance for SNANs, ensuring an optimal balance between variance and bias. We showed that astrocytes improve network performance (depending on their topology within the network) and facilitate stable neuronal spiking.

**4.1 Toward the Development of SNANs for Deep Learning.** Furthermore, we presented a hybrid network of spiking networks and machine learning, derived from the baseline SNN by Diehl and Cook (2015), where the SNAN generates a spiking pattern specific to the input class and the k-NN classifier predicts the output class. Diehl and Cook (2015) reported a classification performance of 82.90% for 100 first layer neurons. With our proposed baseline SNN, we acquired a relative value equal to 82.46%. Our analysis suggests that even though the baseline SNN has the highest accuracy, it displays underfitting caused by insufficient training iterations shown by the difference in training and validation accuracies (see Figures 9 to 11). For the SNN parameters of Diehl and Cook, the maximum spiking input frequency is approximately 63 Hz, while in our proposed architecture,
the maximum input frequency is about 32 Hz, because spiking frequencies above 32 Hz cause oversaturation of astrocytic $Ca^{2+}$ dynamics, preventing astrocytic activations. Consequently, synaptic weight updates are slower. However, there was no validation set in Diehl and Cook's (2015) study. We reported a maximum test accuracy of 75.28% for the SNAN, displaying optimal variance and bias trade-off.

Nazari et al. (2019) showed a 77.35% accuracy in digit classification using a cortical neuron-astrocyte network that was trained on the MNIST data set; this was in contrast to our proposed network. Their extensive architecture had a significant number of fully connected astrocytes (1,501,674) in addition to leaky integrate-and-fire neurons (5,000), which resulted in a marginally superior performance of 2.07% over our network. However, our proposed network featured a simplified configuration of only 100 astrocytes (784 + 100 IF neurons) and showed comparable results while benefiting from a more simplified design. In a separate work by Nazari and Faez (2019), the incorporation of 2500 astrocytes forming tripartite synapses with pyramidal neurons (2500) and interneurons (2500) improved the network accuracy. These astrocytes played a crucial role in mitigating errors caused by neurons being in an always-firing state (Nazari et al., 2019). Determining the optimal count of astrocytic and tripartite synaptic connections is an important factor in designing the SNAN. Our study shows that augmenting the number of astrocytic connections does not necessarily lead to a corresponding improvement in network performance, as shown in the accuracy curves of networks with 50% and 60% tripartite connections (see Figures 9 to 11). Although astrocytes play a critical role in preventing neuronal overexcitation, it is important to note that they can also inadvertently inhibit neuronal excitation, potentially resulting in slower learning.

Rastogi et al. (2021) recently demonstrated SNAN designs using the MNIST data set, where astrocytes modulate the presynaptic neuron release probability during faults when synaptic learning is stuck at zero. In this case, the astrocytes indirectly modulate synaptic activities. In our proposed network, astrocytes act as excitatory inputs to the postsynaptic neurons with STDP ability. To our knowledge, the proposed neuron-astrocyte network is one of the first attempts to identify the potential of astrocytes in image classification using the standard MNIST data set. Therefore, this study can serve as a baseline for evaluating future studies on astrocyte implementation in SNNs.

## 4.2 Astrocytes Improve Network Performance.
Our results suggest that astrocytes influence and improve three attributes of the spiking network: (1) faster learning, (2) variance and bias trade-off, and (3) simplified network architecture. During training, the SNANs displayed higher accuracy than SNN (see Figure 8), indicative of the ability of the network to stabilize its spiking activity quickly. Moreover, the SNANs achieved peak accuracy with fewer input set presentations (see Table 4). These activities
support the idea that astrocyte-mediated neuronal spiking aids in faster
learning. For traditional SNN and artificial networks, increased neuronal
connectivity equates to fast and precise learning. Therefore, the astrocytes
and the corresponding tripartite synapse promote efficient learning by
providing additional synaptic connections in parallel with the neuron-to-
neuron couplings, thus strengthening the communication strength between
neuronal layers. Also, as the neuron spikes more frequently, the spiking
threshold increases, thus requiring more input eventually. Astrocytes there-
fore provide additional input for equalizing neuronal firing rate.

Second, networks with astrocytes balanced variance and bias, as shown
in comparing the training and validation accuracies (see Figures 9 to 11).
The SNAN3 with 30% tripartite synaptic connections exhibits optimal
variance and bias trade-off, with higher training accuracy than SNN and
comparable validation accuracy. These metrics ensure that the spiking pat-
terns from the validation set matched the spiking patterns produced during
training. The SNN displays underfitting, exhibited by a higher validation
accuracy than training accuracy, conveying that the trained network has dif-
ficulty generalizing new data. Defining the SNAN architecture is a highly
iterative process due to the heterogeneity of the astrocytes combined with
the stochastic spiking activity. Therefore, the number of astrocyte-to-neuron
synapses must be sufficient to influence the postsynaptic neuron spiking
and low enough to avoid neuronal overexcitation resulting in erratic neu-
ronal spiking patterns. The disruption of astrocyte function results in a dis-
turbance in the interaction between neurons and astrocytes. Consequently,
this imbalance affects synaptic transmission and leads to an elevated level
of excitation within the neuronal population (Nazari & Faez, 2019). These
effects are noticeable in the networks with more than 30% tripartite connec-
tions (see Figure 10), whose validation accuracy suddenly drop to minimum
levels.

Finally, we have developed a SNAN where astrocytes are point processes
with simpler models while retaining the essential astrocytic dynamics. In-
terestingly, astrocytes can simplify the network architecture. An SNN can
improve learning speed and accuracy by increasing synaptic connectivity,
with optimal results typically achieved through dense synaptic connec-
tions. When dense connectivity is insufficient, another approach is to in-
crease the number of postsynaptic neurons. For example, Saunders et al.
(2018) demonstrated that expanding a network from 100 to 400 postsy-
naptic neurons increased training accuracy from approximately 67% to
70%. However, in this configuration, a network with 400 neurons requires
316,600 synaptic connections to achieve this improvement. In contrast,
the proposed SNAN3 model, which incorporates 30% tripartite (astrocyte-
integrated) synapses, achieved 75.28% accuracy with significantly fewer
connections: 78,400 direct synaptic connections plus an additional 47,040
connections for neuron-to-astrocyte and astrocyte-to-neuron interactions.
This model requires only 200 active processing units (100 neurons and 100
astrocytes) to achieve these results, underscoring the computational effi-
ciency and accuracy benefits of including astrocytes as processing units
rather than merely adding more neurons. Recently, Kozachkov, Slotine, &
Krotov (2023) presented simulation results showing that neuron-astrocyte
models outperform neuron-only models in associative memory tasks. Their
findings indicate that astrocytes actively contribute to memory storage,
with memories distributed across both astrocytes and synapses, rather than
being confined to synapses alone. This suggests that enhancing neuron-
astrocyte connectivity may be a more effective strategy for boosting net-
work performance than merely increasing neuron numbers.

Indeed, the SNAN architecture demonstrated lower accuracy compared
to previous SNN models, which can be attributed to several key factors:

*   **Lower input frequency:** Higher input frequencies generally accelerate learning curves, as excitatory synaptic inputs are updated more frequently, promoting faster adaptation. In our SNAN model, however, we employed a low Poisson spiking frequency of 32 Hz. Increasing input frequency beyond this level leads to overexcitation, which can trigger oversaturation in the Ca²⁺ level. Biologically, overexcitation risks excitotoxicity, a condition harmful to both neurons and astrocytes (Hassel & Dingledine, 2012). In our simulations, frequencies above 35 Hz resulted in Ca²⁺ oversaturation within astrocytes, disrupting their capacity to modulate synaptic transmission effectively. As a result, a slower input spiking rate, while biologically realistic, inevitably leads to slower learning.

*   **Unsupervised learning approach:** The SNAN model employs an unsupervised learning mechanism, meaning there is no feedback to guide or correct learning trajectories, which naturally slows the training process (Diehl & Cook, 2015). The lack of external correction or reinforcement, as is typical in unsupervised models, places limitations on the rate of accuracy improvement.

*   **Simplified model design with a biological focus:** In developing SNAN, we prioritized biological realism in astrocyte and synapse dynamics over extensive parameter tuning or additional layers of complexity. Our objective was to demonstrate the computational potential of astrocytes without the added intricacies of parameter optimization or multiple network layers, which could enhance accuracy but fall outside the goals of this exploratory study.

**4.3 Neuromorphic Neuron-Astrocyte Networks.** The accuracy of the SNAN model reflects the trade-offs we made to maintain biological relevance and simplicity in this initial exploration of astrocyte-based computation. Given these constraints, we propose to simulate the network using neuromorphic systems, as they inherently align with the biological processing style and energy efficiency of neuron-astrocyte interactions.
One of the main challenges we faced during network design is the simulation costs, where simulations are restricted to series computation rather than in parallel. The additional astrocytic components plus the neuron-astrocyte couplings require more extended simulation and computationally heavy programs. Instead of using the Li-Rinzel model, we opted for the Postnov model and simplified the corresponding code using the Brian2 simulator with high-performance computing resources. Researchers can also design and implement SNAN using neuromorphic systems or hardware to solve the series computation restrictions. Two projects aim to implement astrocytes in neuromorphic chips. One project, known as BioRC, is conducted by the University of Southern California (Irizarry-Valle & Parker, 2014). The other project is a collaborative effort between the University of Tehran and the University of Kermanshah in Iran implemented on field programmable gate arrays (FPGAs; Faramarzi et al., 2019). Additionally, the NASA-IMEDIR group from the University of A Coruña has developed an artificial neuron-glia network (ANGN), which combines artificial neurons and artificial astrocytes as different types of processing elements (Pastur-Romay et al., 2016). Kozachkov, Kastanenka, & Krotov (2023) demonstrated that neuron-astrocyte networks could be modeled on transformer architectures, utilizing astrocytes as memory buffers to improve neuromorphic systems. By emulating the attention mechanism central to transformers, systems with astrocyte-like processing units could enable more adaptable and efficient computation, enhancing performance in tasks such as visual recognition.

This study is one of the initial attempts to incorporate astrocytes in the context of third-generation neural networks. Due to computational constraints, each configuration of the proposed neuron-astrocyte network was simulated for only 25 epochs. By advancing to hardware implementation, we will be able to extend the training process and potentially address the issue of underfitting. Therefore, we can conduct a comprehensive performance comparison using various data sets.

In summary, we demonstrate the viability of integrating astrocytes into artificial networks in conjunction with neurons, albeit with slower dynamics. In this study, we developed an astrocyte-modulated spiking neuron network optimized for image recognition. Mimicking their counterparts in biological systems, astrocytes can enhance the performance of spiking networks. However, determining optimal network performance comes with a caveat: it is crucial to find the right balance in the number of tripartite synaptic connections, which is critical for stabilizing network activity and preventing neuronal overexcitation while avoiding the risk of underexcitation. In this proof of concept, therefore, we demonstrate the potential of astrocytes to refine neural network dynamics and functionality. Due to computational constraints, we limited our study to the MNIST data set. Researchers can extend the investigations by training the network on larger
data sets. Additionally, exploring its applicability within neuromorphic sys-
tems could facilitate faster computation and yield improved results.

Acknowledgments

This work was partially funded by the European Regional Development
Fund and by Junta de Extremadura (IB20040), and the Ministerio de Ciencia
e Innovación through grant PID2019-110315RB-I00 funded by MCIN/AEI/
10.13039/501100011033 (APRISA).

Calculations were also performed using HPC resources from the Direc-
tion du Numérique—Centre de Calcul de l'Université de Bourgogne.

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