## 3 Proposed SNNtrainer3D
### User-Friendly GUI and Dynamic Architecture Editing
More precisely, it is described by the differential Equation (1) [27]:  
\[\tau_{m}\ \frac{dV}{dt}\ =-(V-V_{rest}\ )+\ RI_{input} \tag{1}\]  
where \(V\) is the membrane potential, \(V_{rest}\) is the resting membrane potential, \(\tau_{m}\) is the membrane time constant (\(RC_{m}\), where \(C_{m}\) is the membrane capacitance). When \(V\) reaches the threshold \(V_{threshold}\), a spike is emitted, and \(V\) is reset to \(V_{rest}\), followed by a refractory period during which the neuron cannot spike. The \"Leaky\" class in the proposed implementation represents the LIF neuron, with the \"beta\" parameter controlling the leakiness of the membrane potential over time. The membrane potential dynamics are updated each time step during the forward pass. [PERSON]'s neuron model, essentially the precursor to the LIF neuron, is a more biologically plausible variant of LIF and represents one of the earliest attempts to model neural activity mathematically. It focuses on the idea that the neuron integrates incoming electrical signals until reaching a threshold, triggering an action potential or spike. The equation for [PERSON]'s model can be considered a more straightforward form of the LIF model without explicitly accounting for the leakiness of the membrane. It integrates input current over time until the membrane potential reaches a threshold. More precisely, the primary form of this integration, without considering the decay (leakiness), is seen in Equation (2) [27]:  
\[\frac{dV}{dt}=I_{input} \tag{2}\]  
where \(V\) is the membrane potential, and \(I_{input}\) is the input current. Upon reaching the threshold potential \(V_{threshold}\), the neuron fires a spike, resetting the membrane potential. In practice, this model directly leads to the formulation of the LIF model by introducing a leak term to account for the membrane's resistance and capacitance. Compared to Equation (1), [PERSON]'s model can be viewed as a foundational approach that accumulates input signalsuntil a spike is generated without the leak term \(-(V-V_{rest})\). This makes it a fundamental stepping stone towards more detailed neuron models like the LIF neuron. The Synaptic Dynamics neuron model, or the 2 nd-order Integrate-and-Fire neuron, extends the LIF by incorporating synaptic conductances. It models the neuron's response to excitatory and inhibitory synaptic inputs, making it more biologically realistic than the LIF. It describes how synapses modulate the strength of connections between neurons over time. More precisely, it includes mechanisms for the temporal evolution of synaptic efficacy following the arrival of a presynaptic spike, as seen in Equation (3) [24]:  
\[\tau_{s}\frac{dS}{dt}=-S+\sum\delta(t-t_{spike}) \tag{3}\]  
where \(S\) is the synaptic variable representing the synapse's strength, \(\tau_{s}\) is the synaptic time constant, and \(\delta\) is the Dirac delta function, representing the arrival times of presynaptic spikes (\(t_{spike}\)). The Alpha neuron model, also known as the Alpha Membrane model, is a more biologically plausible variant that models the neuron's membrane potential as a filtered version of the input current using an alpha function kernel. It captures the temporal dynamics of synaptic inputs more accurately than the LIF. More precisely, it provides a more detailed description of synaptic currents by characterizing them with an alpha function shape, reflecting the rise and fall of postsynaptic currents over time, as seen in Equation (4) [27]:  
\[I_{synaptic}(t)=\frac{t}{\tau_{synaptic}}e^{1-\frac{t}{\tau_{synaptic}}}\cdot I _{peak} \tag{4}\]  
where \(I_{synaptic}(t)\) is the synaptic current at the time \(t\), \(\tau_{synaptic}\) is the synaptic time constant, controlling the rise and fall of the current and \(I_{peak}\) represents the peak current. Each of these neuron models contributes to the complexity and functionality of SNNs by offering different perspectives on neuron and synapse behavior.