Bipolar Leaky Integrate-and-Fire (BiLIF) Models

• BiLIF is a class of leaky threshold-process models with symmetric positive and negative thresholds that produce signed spike trains capturing both excitation and inhibition.
• BiLIF encompasses variants such as signed-threshold encoding, generalized sampling, and hardware-inspired designs, each impacting reconstruction methods and sparsity properties.
• Operator-theoretic inversion and quantization analyses, alongside evolving hardware realizations, guide practical implementations and clarify challenges in achieving genuine bipolar neural computation.

Bipolar Leaky Integrate-and-Fire (BiLIF) denotes a class of leaky threshold-process models in which a leaky accumulated state is tested against thresholds of both signs, producing a signed spike train rather than a purely positive event sequence. In the supplied literature, this notion appears in three closely related but non-identical forms: as a mathematical encoder with spike signs $\pm 1$ or $\pm \vartheta$; as an operator-theoretic generalized sampler whose inverse problem can be studied with POCS, pseudo-inverses, and approximate bandwidth assumptions; and, more loosely, as a hardware design target in which integrate, leak, threshold, and reset are realized by magnetic or semiconductor state variables without necessarily implementing dual signed thresholds. A recurrent terminological issue is that “bipolar” can refer either to signed neural computation or to bipolar transistor action in device physics, and those meanings should not be conflated (Thao et al., 2022, Moser et al., 2024, Carbajal et al., 2024, Kamal et al., 2019).

1. Scope and conceptual variants

BiLIF is not represented by a single universally adopted formalism. In the strictest sense, it is a one-state LIF encoder with symmetric positive and negative threshold crossings, so that event polarity preserves the sign of the accumulated charge. This is the sense made explicit by signed event models with $\epsilon_n \in \{\pm 1\}$ or $q_j \in \{-1,+1\}$ for real-valued signals. In a broader signal-processing sense, BiLIF includes signed threshold-based time encoders whose output can be treated as generalized nonuniform samples of the input. In a still broader hardware sense, the term is often invoked when a physical device exhibits integrate, leak, threshold, and reset, even if the reported implementation is actually unipolar and only indirectly relevant to signed operation (Thao et al., 2022, Carbajal et al., 2024, Lone et al., 2022).

Variant	Core feature	Representative sources
Signed-threshold BiLIF	One leaky state, thresholds of both signs, signed spikes	(Thao et al., 2022, Moser et al., 2024, Carbajal et al., 2024)
Generalized-sampling BiLIF	Spike times and signs define linear measurement constraints	(Thao et al., 2022, Carbajal et al., 2024)
Embedded threshold reduction	A transformed amplitude variable inherits threshold dynamics from a higher-dimensional neuron	(Yamakou et al., 2018)
Device-level LIF with partial BiLIF relevance	Physical integrate/leak/threshold substrates without explicit signed dual-threshold firing	(Kamal et al., 2019, Lone et al., 2022, Sekh et al., 2024)

A central conceptual point is that bipolarity does not require two separate integrators or two physically distinct channels. The recent mathematical literature shows that a single leaky state with a symmetric threshold rule already yields a bipolar event stream. Conversely, several hardware papers provide convincing realizations of integrate and leak while stopping short of a full signed-threshold BiLIF abstraction. This distinction is essential when comparing formal theory, decoding algorithms, and device claims (Thao et al., 2022, Sekh et al., 2024).

2. Canonical signed-threshold formulations

A standard formalization appears in the bandlimited reconstruction framework, where the LIF output is the impulse train

$p(t)=\sum_{n=1}^N \epsilon_n\delta(t-t_n),$

with $\epsilon_n\in\{\pm1\}$. The spike times are defined recursively from $t_0=0$ by

$$t_n:=\min\Big\{t>t_{n-1}:\Big|\int_{t_{n-1}}^t e^{-\alpha(t-s)}\big(x(s)+c\big)\,ds\Big| =\theta\Big\},$$

and at the firing time

$$\int_{t_{n-1}}^{t_n} e^{-\alpha(t_n-s)}\big(x(s)+c\big)\,ds=\epsilon_n\theta.$$

Here $\alpha\ge 0$ is the leakage, $\pm \vartheta$0 is a bias, and $\pm \vartheta$1 is the threshold. The unipolar case is obtained by restricting to $\pm \vartheta$2, in which case $\pm \vartheta$3 always; the bipolar case is therefore already built into the general model rather than being an external extension (Thao et al., 2022).

A closely related finite-window formulation defines firing times $\pm \vartheta$4 by

$\pm \vartheta$5

with firing phases

$\pm \vartheta$6

For real-valued signals, $\pm \vartheta$7, and the output is the signed spike train

$\pm \vartheta$8

This formulation makes the polarity variable explicit and treats the encoder as a signed time-encoding machine on a finite observation window rather than as a purely neuronal membrane-reset ODE (Carbajal et al., 2024).

A more general continuous-time definition includes refractory time and reset impulses. For input $\pm \vartheta$9 that is almost everywhere bounded, locally integrable, and may be superimposed with locally finitely many Dirac impulses, the spike train is

$\epsilon_n \in \{\pm 1\}$0

with spike times

$\epsilon_n \in \{\pm 1\}$1

In the zero-refractory subtraction-reset case, the discrete-time recursion becomes

$\epsilon_n \in \{\pm 1\}$2

with $\epsilon_n \in \{\pm 1\}$3 the threshold quantizer. This formulation is already signed, since positive and negative threshold crossings yield positive and negative spikes, and the special case $\epsilon_n \in \{\pm 1\}$4 reduces to IF (Moser et al., 2024).

3. Inverse problems, generalized sampling, and reconstruction

Once the event times and signs are known, BiLIF decoding can be reframed as inversion of a linear sampling operator. In the POCS-based reconstruction theory for bandlimited signals, each interspike interval $\epsilon_n \in \{\pm 1\}$5 defines a kernel

$\epsilon_n \in \{\pm 1\}$6

and the firing relation becomes

$\epsilon_n \in \{\pm 1\}$7

where

$\epsilon_n \in \{\pm 1\}$8

For bandlimited inputs, the bandlimited kernels $\epsilon_n \in \{\pm 1\}$9 define the sampling operator

$q_j \in \{-1,+1\}$0

Reconstruction is then posed as solving $q_j \in \{-1,+1\}$1. The POCS operator $q_j \in \{-1,+1\}$2 converges to the jointly bandlimited-and-consistent estimate, and the limit equals the weighted pseudo-inverse solution $q_j \in \{-1,+1\}$3. Under injectivity, recovery is perfect; under incomplete sampling, the limit is the minimum-norm feasible solution. In the reported time-quantization experiments, the weighted pseudo-inverse $q_j \in \{-1,+1\}$4 outperformed the Euclidean pseudo-inverse $q_j \in \{-1,+1\}$5 by roughly $q_j \in \{-1,+1\}$6–$q_j \in \{-1,+1\}$7 dB MSE across tested time resolutions, and POCS was about $q_j \in \{-1,+1\}$8 dB better than Lazar in the stated oversampling experiment. The paper also gives an event-domain discrete implementation

$q_j \in \{-1,+1\}$9

so the iterative computation occurs in event index space rather than at the Nyquist rate (Thao et al., 2022).

A complementary theory replaces exact bandlimitation by a model-agnostic approximate bandwidth condition,

$p(t)=\sum_{n=1}^N \epsilon_n\delta(t-t_n),$0

with main interest in the case $p(t)=\sum_{n=1}^N \epsilon_n\delta(t-t_n),$1. The decoder reconstructs a low-pass approximation from approximate spike times, exact signs, and a nominal leak $p(t)=\sum_{n=1}^N \epsilon_n\delta(t-t_n),$2. The analysis explicitly incorporates leakage uncertainty

$p(t)=\sum_{n=1}^N \epsilon_n\delta(t-t_n),$3

and spike-time uncertainty

$p(t)=\sum_{n=1}^N \epsilon_n\delta(t-t_n),$4

Timing discrepancy is interpreted through Wasserstein-1 distance, and the reconstruction guarantee on an interior inference window scales with $p(t)=\sum_{n=1}^N \epsilon_n\delta(t-t_n),$5, bandwidth, leak mismatch, timing error, and spike count. In the regime $p(t)=\sum_{n=1}^N \epsilon_n\delta(t-t_n),$6, the uncertainty terms remain of the same order as the intrinsic threshold-limited accuracy. This places signed LIF reconstruction on a robust footing for finite windows, approximate bandwidth classes, and imperfect hardware calibration (Carbajal et al., 2024).

These results are significant for BiLIF because the sign of an event does not destroy linear inversion structure. In the generalized-sampling view, polarity changes the measurement scalar and, in more elaborate variants, may change the kernel family, but the decoder remains an operator inversion problem rather than an opaque nonlinear post hoc procedure. This is the most important bridge between signed threshold models and practical reconstruction algorithms (Thao et al., 2022, Carbajal et al., 2024).

4. Quantization, sparsity, and approximation guarantees

A second major line of work interprets LIF as a threshold-based analog-to-spike quantizer controlled by the weighted Alexiewicz norm

$p(t)=\sum_{n=1}^N \epsilon_n\delta(t-t_n),$7

For the discrete case,

$p(t)=\sum_{n=1}^N \epsilon_n\delta(t-t_n),$8

Within this framework, LIF satisfies the quantization bound

$p(t)=\sum_{n=1}^N \epsilon_n\delta(t-t_n),$9

and is idempotent:

$\epsilon_n\in\{\pm1\}$0

The output spike amplitudes lie in $\epsilon_n\in\{\pm1\}$1, and the formulation explicitly accommodates positive and negative threshold crossings, so the theory is directly compatible with signed BiLIF encoders (Moser et al., 2024).

The same framework yields sparsity results. For any integrable $\epsilon_n\in\{\pm1\}$2 bounded except at finitely many Dirac impulses,

$\epsilon_n\in\{\pm1\}$3

Thus the LIF output is no less sparse than the input in $\epsilon_n\in\{\pm1\}$4 norm, although it need not be maximally sparse among all admissible approximants. By contrast, in the zero-leak case one has the exact decomposition

$\epsilon_n\in\{\pm1\}$5

and IF is maximally sparse among spike trains inside the open Alexiewicz ball of radius $\epsilon_n\in\{\pm1\}$6. The paper explicitly shows that this extremal sparsity property does not generally extend to leaky LIF: counterexamples exist in which LIF emits many alternating-sign spikes while another admissible spike train in the same Alexiewicz ball contains only a single spike. The reported empirical observation is that for nonzero leakage and smoother signals, extremal sparsity still appears with very high probability, stated as above $\epsilon_n\in\{\pm1\}$7, but that is not elevated to a general theorem (Moser et al., 2024).

For BiLIF, the importance of this theory is twofold. First, signed spikes are not an auxiliary notation; they are part of the core quantization mechanism. Second, leak fundamentally changes the optimality structure: exact extremal sparsity is available for IF, but not generically for LIF. Any BiLIF claim about “optimal sparse signed encoding” therefore needs to specify whether it is referring to the leakage-free limit, to a theorem in the Alexiewicz geometry, or to an empirical regularity rather than a universal result (Moser et al., 2024).

5. Reduction from biophysical dynamics and network-level generalizations

BiLIF is often introduced as an abstract encoder, but related work on conductance-style neuron models shows how LIF-like threshold processes can emerge from higher-dimensional stochastic dynamics. In the excitable stochastic FitzHugh–Nagumo system,

$\epsilon_n\in\{\pm1\}$8

the excitable regime is defined by a unique stable fixed point, not by coexistence of a fixed point and a limit cycle. After proving existence of a global random pullback attractor, the dynamics near the deterministic equilibrium are linearized, transformed to a damped rotation, and averaged to a radial Ornstein–Uhlenbeck-type process whose threshold crossing with reset acts as an embedded LIF model. The interspike interval statistics of this radial process numerically match those of the original stochastic FitzHugh–Nagumo system. The paper does not construct a two-sided threshold BiLIF model, but it shows that a higher-dimensional excitable stochastic neuron can reduce to a one-dimensional threshold process on a transformed amplitude variable, with the firing rule inherited from the geometry rather than imposed ad hoc. This suggests a principled route by which future BiLIF models could arise from signed or symmetry-reduced coordinates of richer excitable systems (Yamakou et al., 2018).

At the network scale, two-dimensional nonlocally coupled LIF systems provide transferable results on localization even though they are explicitly unipolar. In a toroidal $\epsilon_n\in\{\pm1\}$9 network of standard LIF neurons with one threshold, one reset value, and optional refractory clamping, bump states consist of active asynchronous regions embedded in a quiescent subthreshold background. The study finds that adding a refractory period can localize otherwise traveling bump states, and that randomly preselected permanently idle nodes can also stabilize their positions. For stationary bumps in the infinite-size limit, the paper derives a self-consistency equation for the mean field. The work contains no bipolar membrane states, dual thresholds, or signed spike semantics; nevertheless, it indicates that refractoriness, structural inactivity, and self-consistent mean-field partition into active and quiescent regions are likely to remain important if a genuinely bipolar extension is constructed (Provata et al., 2024).

These results temper a common simplification. BiLIF is not only an encoder-level matter of adding negative spikes. In more realistic dynamical settings, threshold geometry, asymptotic attractors, refractoriness, and spatial coupling can all determine whether a signed threshold description is faithful, localized, or dynamically stable (Yamakou et al., 2018, Provata et al., 2024).

6. Hardware realizations and interpretive boundaries

Several device papers realize LIF behavior in physical substrates and are frequently invoked in discussions of BiLIF, but they differ sharply in what “bipolar” means. The silicon L-BIMOS neuron is the clearest cautionary case. Its operation relies on impact ionization, floating-body charging, and parasitic BJT positive feedback in an L-shaped gate bipolar impact-ionization MOSFET. The authors report reduced breakdown voltage $t_0=0$0 V, firing threshold $t_0=0$1 V, current threshold $t_0=0$2, minimum spike energy $t_0=0$3 pJ, and GHz-range spiking when the drain is biased at $t_0=0$4 V; the results are obtained by $t_0=0$5-D TCAD simulation. Here, however, “bipolar” refers to bipolar transistor action in the device physics, not to positive and negative membrane integration or dual-polarity firing. The reset is externally imposed by driving $t_0=0$6 V, so the neuron is LIF-like but not computationally bipolar in the BiLIF sense (Kamal et al., 2019).

Spintronic and magnetic devices provide more direct analog state variables, but again with only partial BiLIF realization. In a patterned CoFeB/MgO/CoFeB/Ta magnetic tunnel junction, the effective neuron state is the skyrmion configuration, especially its diameter, average $t_0=0$7, and resulting MTJ output voltage. The device realizes integrate and leak through voltage-controlled skyrmion diameter modulation and relaxation, with gradual neuromorphic switching occurring for $t_0=0$8. The paper explicitly states that it does not directly demonstrate a bipolar LIF neuron; rather, it provides a concrete physical substrate for integrate, leak, and threshold, plus only a partial foundation for signed or bidirectional state updates. An SNN built from these skyrmion-based LIF neurons showed the capability of classifying images from the MNIST dataset (Lone et al., 2022).

A second magnetic example uses spin-orbit torque-driven domain-wall motion for integration and synthetic antiferromagnetic coupling for leak and self-reset. The domain-wall position, observed through anomalous Hall voltage, acts as the internal analog state. The maximum domain-wall velocity during the leaky process is reported as $t_0=0$9, and repeated integrate/reset cycles are demonstrated. Yet this device remains effectively unipolar: the reported operation is accumulation toward one firing boundary followed by passive relaxation, without a demonstrated signed membrane state, dual threshold, or positive-versus-negative output spikes. Its relevance to BiLIF is therefore architectural rather than formal: it supplies a practical physical leak mechanism and a movable analog state variable that could support a bipolar extension if the geometry and sensing were redesigned (Sekh et al., 2024).

Taken together, the device literature demonstrates that integrate, leak, threshold, and reset can be mapped onto floating-body charge, skyrmion size, domain-wall position, MTJ resistance, or Hall voltage. It does not, in the supplied papers, yet present a complete hardware BiLIF neuron with explicit symmetric signed thresholds and signed event semantics. The most persistent misconception is therefore terminological: a hardware LIF neuron with an analog state, or a device whose physics is “bipolar,” is not automatically a BiLIF system in the computational sense. In the current literature, the most mature BiLIF foundations are mathematical and operator-theoretic, whereas hardware implementations mostly provide constituent mechanisms rather than a finished signed dual-threshold architecture (Kamal et al., 2019, Lone et al., 2022, Sekh et al., 2024).