The considered datasets were divided into two subsets: 90.00% for training and 10.00% for testing, ensuring that the models had sufficient data to learn while retaining a separate evaluation set. Accuracy was used the primary metric for evaluating classification performance. All computations were provided on Intel(R) Core(TM) i7-14700F, 2.10 GHz.

Comparison of commonly used learning algorithms across neuron models (rows: LIF, LIF input + perceptron hidden/output, Perceptron, Meta, LB) and input processes (columns: Bernoulli, Markov, Poisson). (a) Accuracy [%]; (b) computation time [min]

First, we consider four spiking neural networks made of LIF, perceptron, meta neurons, LB neuron model as well as the hybrid network, in which input layer was construed by LIF neurons, while the hidden and output layers were made of perceprtrons, respectively. In and Fig. 2 and Table 5 in Appendix the influence of neuron model on commonly used learning algorithm, i.e. BP learning algorithm was presented. It turned out that all cases gave accuracies above 90.00%, except for the use of a network composed of the LIF model to the Poisson source, however, the computation times differed significantly. The application of the compared to a neural network composed of LIF neurons, the use of the meta nueron model in the case of BP learning algortihms gives higher accuracy in comparable computation time. Surprisingly, for input data in the form of the Bernoulli process (actually the simplest data set), a neural network consisting of 512 LIF neuron in input layer and 512 perceptrons in hidden and output layers was needed to achieve high accuracy. In comparison, architectures composed of the remaining neurons models required only 64 neurons per layer. Moreover, in the case of the Poisson process, the network based on LIF neurons also required four times more epochs to achieves lower accuracy than in other cases.

Bio-inspired algorithms across neuron models and input processes. Panels (a,c,e) show accuracy (%); panels (b,d,f) show computation time (min). Rows of each heatmap correspond to neuron models (LIF, LB), columns to algorithms (Tempotron, SDSP, STDP, BAL)

Figure 3 and Table 6 in Appendix show the influence of neuron model on bio-inspired learning algorithms like tempotron learning rule, Bio-inspired Active Learning (BAL), STDP, and SDSP. We consider the same architectures as in Table 5. The results obtained show that involving biologically inspired learning algorithms in the process of training a neural network allows for significantly shortened computation time, especially when the neural network is built from LIF, meta, and LB neuron models. The meta and LB neurons models consistently demonstrated high efficiency in handling Bernoulli processes across different bio-inspired learning algorithms. It achieved a balance between accuracy and computational time, making it well-suited for time-sensitive applications requiring moderate accuracy. The first approach demonstrates high efficacy for Bernoulli sequences, where the independence between events reduces the necessity for complex temporal integration. The Levy-Baxter model, although slightly slower, consistently achieved perfect accuracy across all Bernoulli datasets. The introduction of perceptrons into neural networks presented trade-offs: while they occasionally improved processing speed, particularly in the BAL and tempotron algorithms, they generally led to reduced accuracy and significantly increased training times, as observed in the case of STDP-based algorithms. The Levy-Baxter neural model exhibited superior accuracy, consistently reaching 99.00-100.00% in all scenarios tested. Despite requiring slightly longer training times compared to simpler neuron models, its adaptability and robustness in BAL scenarios make it a strong candidate for accuracy-critical applications. The application of the tempotron learning algorithm enables high-accuracy computations within optimal time constraints. Furthermore, meta-neurons demonstrated an accuracy range of 90 to 100% when implemented in SNNs with 32 to 64 neurons, while other neuron models typically required 128 neurons or more to achieve comparable performance.

The analysis of neuron models under BAL, tempotron, and STDP learning algorithms for Markov sequences revealed distinct performance trends, emphasizing the trade-offs between accuracy and computational efficiency. The perceptron model consistently excelled, achieving perfect accuracy (i.e., 100.00%) in BP and STDP scenarios, demonstrating its effectiveness in capturing Markovian dependencies. Hybrid models integrating LIF neurons with perceptrons exhibited significant potential under BP and bio-inspired learning algorithms, balancing low computational cost with high accuracy. However, these models showed inefficiencies under SDPD, where accuracy declined to 83.00%. The biologically inspired LB model demonstrated exceptional robustness, achieving near-perfect accuracy (93.75–100.00%) across tasks. However, its substantially longer runtimes make it more suitable for precision-critical applications rather than time-sensitive computations. Conversely, meta neurons provided a compelling balance between accuracy and efficiency, maintaining high accuracy while requiring only 32-64 neurons per layer, significantly reducing computational overhead compared to alternative neuron models.

For datasets containing Poisson processes, the perceptron-based neural network model consistently demonstrated the highest efficiency and effectiveness, achieving perfect accuracy (i.e. 100.00%) in minimal time under the tempotron learning algorithm and SDPS. However, in the case of BAL, the computational time increased by an order of magnitude while maintaining the same accuracy. Hybrid models, such as those combining LIF neurons with perceptrons, exhibited significantly lower accuracy, ranging from 68.00% to 89.00%, but with relatively low computational costs. While this approach benefits from step dynamics and linear decision boundaries, it requires careful tuning to prevent inefficiencies, particularly with increased learning periods or larger network sizes. Under the SDSP algorithm, a configuration with 16 neurons and 10 epochs achieved an accuracy of 68.00% with a runtime of 28.1 seconds. Similarly, under the STDP algorithm, the same configuration yielded the same accuracy (i.e. 68.00%) but with a significantly reduced runtime of 2.7 seconds, highlighting the computational efficiency of STDP. In contrast, applying the BAL algorithm to a larger network configuration (64 neurons, 20 epochs) resulted in a significantly higher accuracy of 88.00%, with a runtime of 1 minute and 48.0 seconds. Biologically inspired models, such as the LB model, exhibited robustness and representational richness, achieving accuracy levels between 91.00% and 99.00% with 64–128 neurons under bio-inspired learning algorithms, albeit at the cost of increased computational time. In turn, metaneuron networks consistently provided high accuracy with low computational costs for 64 neurons per layer across all bio-inspired learning algorithms, except for tempotron. In the case of the tempotron learning algorithm, achieving comparable accuracy required 128 neurons per layer. Nevertheless, the computational cost remained an order of magnitude lower than in previous cases, reinforcing the efficiency of metaneuron networks.

The optimal neuron model and learning algorithm depend on the application’s accuracy and efficiency requirements. LB models excel in accuracy-critical tasks, while LIF-based architectures with BAL provide efficient solutions. Hybrid models offer a promising middle ground, performing well when paired with appropriate learning algorithms (Fig. 4).

Results for the LIF and LB neuron models across three input processes: Bernoulli, Markov, Poisson. Rows correspond to input processes. The left column reports accuracy (%) versus the number of neurons per layer (log\(_2\) scale) for a commonly applied learning rule (e.g., BP). The right column shows time–accuracy trade-offs (log-scaled time). Bubble area is proportional to neuron count (\(\propto N\)), and color encodes the learning algorithm. Colors and size mapping are kept consistent across panels for direct comparison

Also taking into account the results obtained in the Paprocki et al. (2024) paper, namely that a large number of neurons in the network does not necessarily lead to significant improvements in transmission efficiency but can enhance the reliability of the system, we examined the influence of the number of neurons in individual layers on accuracy and computation time. Figures 5 and 6 present time-accuracy trade-off for neuron model with input process. Each marker corresponds to one network configuration trained for 10 epochs. Color encodes the learning algorithm (BP, BAL, SDSP, STDP/STPD, Tempotron). The x-axis shows wall-clock training time in seconds on a logarithmic scale (tick marks are powers of ten), and the y-axis reports classification accuracy (%). Bubble size encodes model scale: the area of each bubble is proportional to the number of neurons per layer (\(\textrm\propto N\)). Consequently, doubling N doubles the bubble area, while the bubble diameter grows only with \(\sqrt\). This area-based encoding preserves perceptual proportionality and avoids over-emphasizing very large models. For orientation, the legends include three reference sizes (16 / 128 / 1024 neurons), and all other sizes are interpolated proportionally. The same color and size mappings are used consistently across figures, enabling direct visual comparison. Points closer to the upper-left corner indicate more favorable configurations on the empirical Pareto front (higher accuracy at lower time). In Tables 7, 8, 9 in Appendix, the influence of numbers of neurons on learning algorithms, taking account BP algorithms and bio-inspired learning algorithms were presented. The neural architecture made by LIF and LB neuron model, respectively, were widely investigated. Variants of neural networks that had 16, 32, 64, 128, 512 and 1024 nuerons in each layer were tested, respectively. In the case of BP learning algorithm (see, Table 7) all computation was provided in the 10 epochs. Both networks composed of LIF and LB neurons achieved high accuracy in the case of data from Bernoulli and Markov processes, however, in the case of the Poisson process, the network model based on LIF neurons achieved a maximum accuracy of 73.50 percent with the number of neurons in the layer also 32. Then, as the number of neurons in the layers increased, the accuracy dropped below 50.00%. The network based on the Levy-Baxter neuron model achieved an accuracy of more than 97. 00% at 64, but the computation time was longer than when using the neural network based on the LIF model. In other cases, there is a visible trend towards an increase in precision as the number of neurons in the layers increases.

Results for the LIF neuron model across three input processes: Bernoulli, Markov, Poisson. (a,c,e) Accuracy (%) against number of neurons (log\(_2\)); (b,d,f) Time–accuracy trade-offs (log time). Bubble area \(\propto N\) and color encodes the algorithm

Results for the LB neuron model across three input processes: Bernoulli, Markov, Poisson. (a,c,e) Accuracy (%) against number of neurons (log\(_2\)); (b,d,f) Time–accuracy trade-offs (log time). Bubble area \(\propto N\) and color encodes the algorithm

In Table 8 we consider the influence of numbers of neurons in neural networks, which consists of LIF neurons on bio-inspired learning algorithms. Calculations were performed for 10 epochs. In the case of Bernoulli process and temporton learning algorithm increasing the number of neurons in the range 16-64 does not result in a significant increase in accuracy. For 128 and 512 neurons in each layer, we obtain an accuracy of over 93.00%, while for 1024 we obtained only 42.50%. When we classify a two-state Markov process, we get such accuracy over 90.00% in the case of 128, 512 and 1024 neurons in layers. However, for the number of neurons in layers 64 it reaches 82.00%. In turn, this learning algorithm does not work for the Poisson process. Only an accuracy greater than 80.00% is achieved for 128 neurons in each layer. The SDSP algorithm allows to achieve high accuracy regardless of the number of neurons in the layers when classifying Bernoulli and Markov processes, while in the case of the Poisson process only when each layer has 64 neurons, i.e. 88.00%. In the case of the BAL algorithm accuracy increases as the number of neurons in layers increases. Surprisingly, the BAL and SDSP learning algorithms can achieve accuracy above 80.00% only for 64 neurons in layers when classifying Poisson processes. The STPD algorithm gives high accuracy for networks composed of a larger number of neurons, however not exceed 128 neurons in each layer. In turn, the computation times for the tempotron, BAL and SDSP learning algorithms are similar for all data. However, in the case of the STDP algorithm, with a large number of neurons in the layers, the computation time is more than twice as high.

In turn, in Table 9 the results obtained for neural network that consist of LB neuron was shown. Calculations were performed for 10 epochs. It turned out that the use of the tempotron learning algorithm gave high accuracy for all considered data, while the neural network architecture had 128, and 512 neurons in each layer. When the number of neurons in the layers was higher, the accuracy dropped below 50.00%. A similar situation occurred when the number of neurons in the layers was smaller. For example, in the case of tempotron learning rule, the small layer sizes (16, 32 neurons) may lack sufficient representational capacity to capture the temporal and probabilistic dependencies in Bernoulli, Markov, and Poisson processes. This results in underfitting, where the model cannot adequately learn the patterns in the data. On the other hand, large layer sizes like 512, 1024 neurons may overfit the data, particularly for simpler processes. Overfitting can occur when the model memorizes specific patterns instead of generalizing, leading to poor performance on test data. A similar situation is with other learning algorithms, only BAL is exception.

To summarize, our representative results show clear, model-dependent trade-offs in both accuracy and runtime. It turned out that for Poisson-distributed inputs, the most optimal configuration is a perceptron trained with the Tempotron rule, achieving 100.00% accuracy in just 8.5 seconds. For Bernoulli data, meta networks trained with BAL or Tempotron offer the best speed-accuracy trade-off under tight computational budgets. If accuracy is prioritized over runtime for Bernoulli inputs, LB models trained with BAL or Tempotron are preferable. In the case of Markov data, LB networks trained with BAL or Tempotron yield the highest accuracy (up to 100.00%) but at a higher computational cost. When balancing speed and performance for Markov input, Tempotron perceptrons remain a strong choice. Overall, model selection should consider both data structure and resource constraints, as optimal configurations vary significantly across input types.

Across datasets, a small set of model-rule pairs consistently dominated, see Table 2. For Poisson inputs, a perceptron trained with the Tempotron rule achieved 100.00% accuracy in 8.5 s with \(64\) neurons and \(10\) epochs, offering the best time-accuracy trade-off. For Bernoulli, meta neurons with BAL reached 99.50% in 6.7 s using \(32\) neurons (fastest high-accuracy option), whereas LB + BAL achieved 100.00% at a modestly higher cost (\(22.2\,\textrm\), \(32\) neurons). For Markov sequences, perceptron + Tempotron attained 100.00% accuracy in 53.6 s with \(128\) neurons. Increasing layer width beyond \(128\) neurons rarely helped and sometimes degraded performance (e.g., LIF on Poisson dropped to \(\le 49\%\) for \(N \ge 64\)). Overall, LB models favor accuracy at higher runtime, meta models strike an efficiency-accuracy balance, and hybrid LIF + perceptron can be competitive on Bernoulli/Markov but lags on Poisson.

On MNIST reference dataset, all architectures except the metaneuron exceed 99.00% accuracy, and the practical differentiator is training cost, see Table 1 in Appendix. A pure LIF network attains the top score (\(\approx 99.90\%\)) but with the longest wall time, reflecting the expense of simulating membrane dynamics and resets. A hybrid LIF + perceptron (LIF input, perceptron hidden/output) delivers almost the same accuracy with a markedly shorter runtime, indicating that precise temporal encoding at the front end is sufficient while a linear/readout layer can close the decision boundary. A plain perceptron provides a very strong, fast baseline (\(\approx 99.30\%\)) on this static vision task. The LB model matches LIF-level accuracy but trains slower due to its stochastic synaptic mechanism. For Poisson inputs, the LB neuron reaches 97.00% already with 64 neurons and 10 epochs, underscoring its robustness to stochastic spike trains.