Large language models (LLMs) continue to face challenges in reliably solving reasoning tasks, particularly tasks that involve precise rule following, as often found in mathematical reasoning tasks.This paper introduces a novel neurosymbolic method that improves LLM reasoning by encoding hidden states into neurosymbolic vectors, enabling problem-solving within a neurosymbolic vector space.The results are decoded and merged with the original hidden state, significantly boosting the model's performance on numerical reasoning tasks.By offloading computation through neurosymbolic representations, this method enhances efficiency, reliability, and interpretability.Our experimental results demonstrate an average of 88.6% lower cross-entropy loss and 15.4 times more problems correctly solved on a suite of mathematical reasoning tasks compared to chain-of-thought prompting and supervised fine-tuning (LoRA), while not hindering the LLM's performance on other tasks.We make our code available at Neurosymbolic LLM 1 .

Improving biological plausibility and functional capacity are two important goals for brain models that connect low-level neural details to high-level behavioral phenomena. We develop a method called "oracle-supervised Neural Engineering Framework" (osNEF) to train biologically-detailed spiking neural networks that realize a variety of cognitively-relevant dynamical systems. Specifically, we train networks to perform computations that are commonly found in cognitive systems (communication, multiplication, harmonic oscillation, and gated working memory) using four distinct neuron models (leaky-integrate-and-fire neurons, Izhikevich neurons, 4-dimensional nonlinear point neurons, and 4-compartment, 6-ion-channel layer-V pyramidal cell reconstructions) connected with various synaptic models (current-based synapses, conductance-based synapses, and voltage-gated synapses). We show that osNEF networks exhibit the target dynamics by accounting for nonlinearities present within the neuron models: performance is comparable across all four systems and all four neuron models, with variance proportional to task and neuron model complexity. We also apply osNEF to build a model of working memory that performs a delayed response task using a combination of pyramidal cells and inhibitory interneurons connected with NMDA and GABA synapses. The baseline performance and forgetting rate of the model are consistent with animal data from delayed match-to-sample tasks (DMTST): we observe a baseline performance of 95% and exponential forgetting with time constant τ = 8.5s, while a recent meta-analysis of DMTST performance across species observed baseline performances of 58 - 99% and exponential forgetting with time constants of τ = 2.4 - 71s. These results demonstrate that osNEF can train functional brain models using biologically-detailed components and open new avenues for investigating the relationship between biophysical mechanisms and functional capabilities.

Recently, a new recurrent neural network (RNN) named the Legendre Memory Unit (LMU) was proposed and shown to achieve state-of-the-art performance on several benchmark datasets. Here we leverage the linear time-invariant (LTI) memory component of the LMU to construct a simplified variant that can be parallelized during training (and yet executed as an RNN during inference), resulting in up to 200 times faster training. We note that our efficient parallelizing scheme is general and is applicable to any deep network whose recurrent components are linear dynamical systems. We demonstrate the improved accuracy of our new architecture compared to the original LMU and a variety of published LSTM and transformer networks across seven benchmarks. For instance, our LMU sets a new state-of-the-art result on psMNIST, and uses half the parameters while outperforming DistilBERT and LSTM models on IMDB sentiment analysis.

Nonlinear interactions in the dendritic tree play a key role in neural computation. Nevertheless, modeling frameworks aimed at the construction of large-scale, functional spiking neural networks, such as the Neural Engineering Framework, tend to assume a linear superposition of postsynaptic currents. In this letter, we present a series of extensions to the Neural Engineering Framework that facilitate the construction of networks incorporating Dale's principle and nonlinear conductance-based synapses. We apply these extensions to a two-compartment LIF neuron that can be seen as a simple model of passive dendritic computation. We show that it is possible to incorporate neuron models with input-dependent nonlinearities into the Neural Engineering Framework without compromising high-level function and that nonlinear postsynaptic currents can be systematically exploited to compute a wide variety of multivariate, band-limited functions, including the Euclidean norm, controlled shunting, and nonnegative multiplication. By avoiding an additional source of spike noise, the function approximation accuracy of a single layer of two-compartment LIF neurons is on a par with or even surpasses that of two-layer spiking neural networks up to a certain target function bandwidth.

While neural networks are highly effective at learning task-relevant representations from data, they typically do not learn representations with the kind of symbolic structure that is hypothesized to support high-level cognitive processes, nor do they naturally model such structures within problem domains that are continuous in space and time. To fill these gaps, this work exploits a method for defining vector representations that bind discrete (symbol-like) entities to points in continuous topological spaces in order to simulate and predict the behavior of a range of dynamical systems. These vector representations are spatial semantic pointers (SSPs), and we demonstrate that they can (1) be used to model dynamical systems involving multiple objects represented in a symbol-like manner and (2) be integrated with deep neural networks to predict the future of physical trajectories. These results help unify what have traditionally appeared to be disparate approaches in machine learning.

Neuromorphic hardware has long promised to provide power advantages by leveraging the kind of event-driven, temporally sparse computation observed in biological neural systems. Only recently, however, has this hardware been developed to a point that allows for general purpose AI programming. In this paper, we provide an overview of tools and methods for building applications that run on neuromorphic computing devices. We then discuss reasons for observed efficiency gains in neuromorphic systems, and provide a concrete illustration of these gains by comparing conventional and neuromorphic implementations of a keyword spotting system trained on the widely used Speech Commands dataset. We show that replacing floating point operations in a conventional neural network with synaptic operations in a spiking neural network results in a roughly 4x energy reduction, with minimal performance loss.

We propose a novel memory cell for recurrent neural networks that dynamically maintains information across long windows of time using relatively few resources. The Legendre Memory Unit~(LMU) is mathematically derived to orthogonalize its continuous-time history -- doing so by solving $d$ coupled ordinary differential equations~(ODEs), whose phase space linearly maps onto sliding windows of time via the Legendre polynomials up to degree $d - 1$. Backpropagation across LMUs outperforms equivalently-sized LSTMs on a chaotic time-series prediction task, improves memory capacity by two orders of magnitude, and significantly reduces training and inference times. LMUs can efficiently handle temporal dependencies spanning $100\text{,}000$ time-steps, converge rapidly, and use few internal state-variables to learn complex functions spanning long windows of time -- exceeding state-of-the-art performance among RNNs on permuted sequential MNIST. These results are due to the network's disposition to learn scale-invariant features independently of step size. Backpropagation through the ODE solver allows each layer to adapt its internal time-step, enabling the network to learn task-relevant time-scales. We demonstrate that LMU memory cells can be implemented using $m$ recurrently-connected Poisson spiking neurons, $\mathcal{O}( m )$ time and memory, with error scaling as $\mathcal{O}( d / \sqrt{m} )$. We discuss implementations of LMUs on analog and digital neuromorphic hardware.

We present a new binding operation, vector-derived transformation binding (VTB), for use in vector symbolic architectures (VSA). The performance of VTB is compared to circular convolution, used in holographic reduced representations (HRRs), in terms of list and stack encoding capacity. A special focus is given to the possibility of a neural implementation by the means of the Neural Engineering Framework (NEF). While the scaling of required neural resources is slightly worse for VTB, it is found to be on par with circular convolution for list encoding and better for encoding of stacks. Furthermore, VTB influences the vector length less, which also benefits a neural implementation. Consequently, we argue that VTB is an improvement over HRRs for neurally implemented VSAs.

Braindrop is the first neuromorphic system designed to be programmed at a high level of abstraction. Previous neuromorphic systems were programmed at the neurosynaptic level and required expert knowledge of the hardware to use. In stark contrast, Braindrop's computations are specified as coupled nonlinear dynamical systems and synthesized to the hardware by an automated procedure. This procedure not only leverages Braindrop's fabric of subthreshold analog circuits as dynamic computational primitives but also compensates for their mismatched and temperature-sensitive responses at the network level. Thus, a clean abstraction is presented to the user. Fabricated in a 28-nm FDSOI process, Braindrop integrates 4096 neurons in 0.65 mm2. Two innovations-sparse encoding through analog spatial convolution and weighted spike-rate summation though digital accumulative thinning-cut digital traffic drastically, reducing the energy Braindrop consumes per equivalent synaptic operation to 381 fJ for typical network configurations.

Low-power, high-speed neural networks are critical for providing deployable embedded AI applications at the edge. We describe an FPGA implementation of Neural Engineering Framework (NEF) networks with online learning that outperforms mobile GPU implementations by an order of magnitude or more. Specifically, we provide an embedded Python-capable PYNQ FPGA implementation supported with a High-Level Synthesis (HLS) workflow that allows sub-millisecond implementation of adaptive neural networks with low-latency, direct I/O access to the physical world. We tune the precision of the different intermediate variables in the code to achieve competitive absolute accuracy against slower and larger floating-point reference designs. The online learning component of the neural network exploits immediate feedback to adjust the network weights to best support a given arithmetic precision. As the space of possible design configurations of such networks is vast and is subject to a target accuracy constraint, we use the Hyperopt hyper-parameter tuning tool instead of manual search to find Pareto optimal designs. Specifically, we are able to generate the optimized designs in under 500 iterations of Vivado HLS before running the complete Vivado place-and-route phase on that subset. For neural network populations of 64-4096 neurons and 1-8 representational dimensions our optimized FPGA implementation generated by Hyperopt has a speedup of 10-484× over a competing cuBLAS implementation on the Jetson TX1 GPU while using 2.4-9.5× less power. Our speedups are a result of HLS-specific reformulation (15× improvement), precision adaptation (4× improvement), and low-latency direct I/O access (1000× improvement).

Researchers building spiking neural networks face the challenge of improving the biological plausibility of their model networks while maintaining the ability to quantitatively characterize network behavior. In this work, we extend the theory behind the neural engineering framework (NEF), a method of building spiking dynamical networks, to permit the use of a broad class of synapse models while maintaining prescribed dynamics up to a given order. This theory improves our understanding of how low-level synaptic properties alter the accuracy of high-level computations in spiking dynamical networks. For completeness, we provide characterizations for both continuous-time (i.e., analog) and discrete-time (i.e., digital) simulations. We demonstrate the utility of these extensions by mapping an optimal delay line onto various spiking dynamical networks using higher-order models of the synapse. We show that these networks nonlinearly encode rolling windows of input history, using a scale invariant representation, with accuracy depending on the frequency content of the input signal. Finally, we reveal that these methods provide a novel explanation of time cell responses during a delay task, which have been observed throughout hippocampus, striatum, and cortex.

The Neural Engineering Framework (NEF) is a theory for mapping computations onto biologically plausible networks of spiking neurons. This theory has been applied to a number of neuromorphic chips. However, within both silicon and real biological systems, synapses exhibit higher-order dynamics and heterogeneity. To date, the NEF has not explicitly addressed how to account for either feature. Here, we analytically extend the NEF to directly harness the dynamics provided by heterogeneous mixed-analog-digital synapses. This theory is successfully validated by simulating two fundamental dynamical systems in Nengo using circuit models validated in SPICE. Thus, our work reveals the potential to engineer robust neuromorphic systems with well-defined high-level behaviour that harness the low-level heterogeneous properties of their physical primitives with millisecond resolution.

The biological brain is a highly plastic system within which the efficacy and structure of synaptic connections are constantly changing in response to internal and external stimuli. While numerous models of this plastic behavior exist at various levels of abstraction, how these mechanisms allow the brain to learn meaningful values is unclear. The Neural Engineering Framework (NEF) is a hypothesis about how large-scale neural systems represent values using populations of spiking neurons, and transform them using functions implemented by the synaptic weights between populations. By exploiting the fact that these connection weight matrices are factorable, we have recently shown that static NEF models can be simulated very efficiently using the SpiNNaker neuromorphic architecture. In this paper, we demonstrate how this approach can be extended to efficiently support both supervised and unsupervised learning rules designed to operate on these factored matrices. We then present a heteroassociative memory architecture built using these learning rules and prove that it is capable of learning a human-scale semantic network. Finally we demonstrate a 100 000 neuron version of this architecture running on the SpiNNaker simulator with a speed-up exceeding 150x when compared to the Nengo reference simulator.

Living organisms are capable of autonomously adapting to dynamically changing environments by receiving inputs from highly specialized sensory organs and elaborating them on the same parallel, power-efficient neural substrate. In this paper we present a prototype for a comprehensive integrated platform that allows replicating principles of neural information processing in real-time. Our system consists of (a) an autonomous mobile robotic platform, (b) on-board actuators and multiple (neuromorphic) sensors, and (c) the SpiNNaker computing system, a configurable neural architecture for exploration of parallel, brain-inspired models. The simulation of neurally inspired perception and reasoning algorithms is performed in real-time by distributed, low-power, low-latency event-driven computing nodes, which can be flexibly configured using C or specialized neural languages such as PyNN and Nengo. We conclude by demonstrating the platform in two experimental scenarios, exhibiting real-world closed loop behavior consisting of environmental perception, reasoning and execution of adequate motor actions.

Brain-inspired, spike-based computation in electronic systems is being investigated for developing alternative, non-conventional computing technologies. The Neural Engineering Framework provides a method for programming these devices to implement computation. In this paper we apply this approach to perform arbitrary mathematical computation using a mixed signal analog/digital neuromorphic multi-neuron VLSI chip. This is achieved by means of a network of spiking neurons with multiple weighted connections. The synaptic weights are stored in a 4-bit on-chip programmable SRAM block. We propose a parallel event-based method for calibrating appropriately the synaptic weights and demonstrate the method by encoding and decoding arbitrary mathematical functions, and by implementing dynamical systems via recurrent connections.

Noise and heterogeneity are both known to benefit neural coding. Stochastic resonance describes how noise, in the form of random fluctuations in a neuron's membrane voltage, can improve neural representations of an input signal. Neuronal heterogeneity refers to variation in any one of a number of neuron parameters and is also known to increase the information content of a population. We explore the interaction between noise and heterogeneity and find that their benefits to neural coding are not independent. Specifically, a neuronal population better represents an input signal when either noise or heterogeneity is added, but adding both does not always improve representation further. To explain this phenomenon, we propose that noise and heterogeneity operate using two shared mechanisms: (1) temporally desynchronizing the firing of neurons in the population and (2) linearizing the response of a population to a stimulus. We first characterize the effects of noise and heterogeneity on the information content of populations of either leaky integrate-and-fire or FitzHugh-Nagumo neurons. We then examine how the mechanisms of desynchronization and linearization produce these effects, and find that they work to distribute information equally across all neurons in the population in terms of both signal timing (desynchronization) and signal amplitude (linearization). Without noise or heterogeneity, all neurons encode the same aspects of the input signal; adding noise or heterogeneity allows neurons to encode complementary aspects of the input signal, thereby increasing information content. The simulations detailed in this letter highlight the importance of heterogeneity and noise in population coding, demonstrate their complex interactions in terms of the information content of neurons, and explain these effects in terms of underlying mechanisms.

In this paper, we review the theoretical and software tools used to construct Spaun, the first (and so far only) brain model capable of performing cognitive tasks. This tool set allowed us to configure 2.5 million simple nonlinear components (neurons) with 60 billion connections between them (synapses) such that the resulting model can perform eight different perceptual, motor, and cognitive tasks. To reverse-engineer the brain in this way, a method is needed that shows how large numbers of simple components, each of which receives thousands of inputs from other components, can be organized to perform the desired computations. We achieve this through the neural engineering framework (NEF), a mathematical theory that provides methods for systematically generating biologically plausible spiking networks to implement nonlinear and linear dynamical systems. On top of this, we propose the semantic pointer architecture (SPA), a hypothesis regarding some aspects of the organization, function, and representational resources used in the mammalian brain. We conclude by discussing Spaun, which is an example model that uses the SPA and is implemented using the NEF. Throughout, we discuss the software tool Neural ENGineering Objects (Nengo), which allows for the synthesis and simulation of neural models efficiently on the scale of Spaun, and provides support for constructing models using the NEF and the SPA. The resulting NEF/SPA/Nengo combination is a general tool set for both evaluating hypotheses about how the brain works, and for building systems that compute particular functions using neuron-like components.

Visuospatial attention produces myriad effects on the activity and selectivity of cortical neurons. Spiking neuron models capable of reproducing a wide variety of these effects remain elusive. We present a model called the Attentional Routing Circuit (ARC) that provides a mechanistic description of selective attentional processing in cortex. The model is described mathematically and implemented at the level of individual spiking neurons, with the computations for performing selective attentional processing being mapped to specific neuron types and laminar circuitry. The model is used to simulate three studies of attention in macaque, and is shown to quantitatively match several observed forms of attentional modulation. Specifically, ARC demonstrates that with shifts of spatial attention, neurons may exhibit shifting and shrinking of receptive fields; increases in responses without changes in selectivity for non-spatial features (i.e. response gain), and; that the effect on contrast-response functions is better explained as a response-gain effect than as contrast-gain. Unlike past models, ARC embodies a single mechanism that unifies the above forms of attentional modulation, is consistent with a wide array of available data, and makes several specific and quantifiable predictions.

Recent papers have shown the possibility to implement large scale neural network models that perform complex algorithms in a biologically realistic way. However, such models have been simulated on architectures unable to perform real-time simulations. In previous work we presented the possibility to simulate simple models in real-time on the SpiNNaker neuromimetic architecture. However, such models were “static”: the algorithm performed was defined at design-time. In this paper we present a novel learning rule, that exploits the peculiarities of the SpiNNaker system, enabling models designed with the Neural Engineering Framework (NEF) to learn transfer functions using a supervised framework. We show that the proposed learning rule, belonging to the Prescribed Error Sensitivity (PES) class, is able to learn, effectively, both linear and non-linear functions.

Simulation of large-scale networks of spiking neurons has become appealing for understanding the computational principles of the nervous system by producing models based on biological evidence. In particular, networks that can assume a variety of (dynamically) stable states have been proposed as the basis for different behavioural and cognitive functions. This work focuses on implementing the Neural Engineering Framework (NEF), a formal method for mapping attractor networks and control-theoretic algorithms to biologically plausible networks of spiking neurons, on the SpiNNaker system, a massive programmable parallel architecture oriented to the simulation of networks of spiking neurons. We describe how to encode and decode analog values to patterns of neural spikes directly on chip. These methods take advantage of the full programmability of the ARM968 cores constituting the processing base of a SpiNNaker node, and exploit the fast Network-on-chip for spike communication. In this paper we focus on the fundamentals of representing, transforming and implementing dynamics in spiking networks. We show real time simulation results demonstrating the NEF principles and discuss advantages, precision and scalability. More generally, the present approach can be used to state and test hypotheses with large-scale spiking neural network models for a range of different cognitive functions and behaviours.

Motor prostheses aim to restore function to disabled patients. Despite compelling proof of concept systems, barriers to clinical translation remain. One challenge is to develop a low-power, fully-implantable system that dissipates only minimal power so as not to damage tissue. To this end, we implemented a Kalman-filter based decoder via a spiking neural network (SNN) and tested it in brain-machine interface (BMI) experiments with a rhesus monkey. The Kalman filter was trained to predict the arm’s velocity and mapped on to the SNN using the Neural Engineer- ing Framework (NEF). A 2,000-neuron embedded Matlab SNN implementation runs in real-time and its closed-loop performance is quite comparable to that of the standard Kalman filter. The success of this closed-loop decoder holds promise for hardware SNN implementations of statistical signal processing algorithms on neuromorphic chips, which may offer power savings necessary to overcome a major obstacle to the successful clinical translation of neural motor prostheses.

This paper re-examines the question of localist vs. distributed neural representations using a biologically realistic framework based on the central notion of neurons having a preferred direction vector. A preferred direction vector captures the general observation that neurons fire most vigorously when the stimulus lies in a particular direction in a represented vector space. This framework has been successful in capturing a wide variety of detailed neural data, although here we focus on cognitive representation. In particular, we describe methods for constructing spiking networks that can represent and manipulate structured, symbol-like representations. In the context of such networks, neuron activities can seem both localist and distributed, depending on the space of inputs being considered. This analysis suggests that claims of a set of neurons being localist or distributed cannot be made sense of without specifying the particular stimulus set used to examine the neurons.

Temporal derivatives are computed by a wide variety of neural circuits, but the problem of performing this computation accurately has received little theoretical study. Here we systematically compare the performance of diverse networks that calculate derivatives using cell-intrinsic adaptation and synaptic depression dynamics, feedforward network dynamics, and recurrent network dynamics. Examples of each type of network are compared by quantifying the errors they introduce into the calculation and their rejection of high-frequency input noise. This comparison is based on both analytical methods and numerical simulations with spiking leaky-integrate-and-fire (LIF) neurons. Both adapting and feedforward-network circuits provide good performance for signals with frequency bands that are well matched to the time constants of postsynaptic current decay and adaptation, respectively. The synaptic depression circuit performs similarly to the adaptation circuit, although strictly speaking, precisely linear differentiation based on synaptic depression is not possible, because depression scales synaptic weights multiplicatively. Feedback circuits introduce greater errors than functionally equivalent feedforward circuits, but they have the useful property that their dynamics are determined by feedback strength. For this reason, these circuits are better suited for calculating the derivatives of signals that evolve on timescales outside the range of membrane dynamics and, possibly, for providing the wide range of timescales needed for precise fractional-order differentiation.

In cortical neural networks, connections from a given neuron are either inhibitory or excitatory but not both. This constraint is often ignored by theoreticians who build models of these systems. There is currently no general solution to the problem of converting such unrealistic network models into biologically plausible models that respect this constraint. We demonstrate a constructive transformation of models that solves this problem for both feedforward and dynamic recurrent networks. The resulting models give a close approximation to the original network functions and temporal dynamics of the system, and they are biologically plausible. More precisely, we identify a general form for the solution to this problem. As a result, we also describe how the precise solution for a given cortical network can be determined empirically.

Extending work in Eliasmith and Anderson (2003), we employ a general framework to construct biologically plausible simulations of the three classes of attractor networks relevant for biological systems: static (point, line, ring, and plane) attractors, cyclic attractors, and chaotic attractors. We discuss these attractors in the context of the neural systems that they have been posited to help explain: eye control, working memory, and head direction; locomotion (specifically swimming); and olfaction, respectively. We then demonstrate how to introduce control into these models. The addition of control shows how attractor networks can be used as subsystems in larger neural systems, demonstrates how a much larger class of networks can be related to attractor networks, and makes it clear how attractor networks can be exploited for various information processing tasks in neurobiological systems.

The properties of Turing's famous ‘universal machine’ has long sustained functionalist intuitions about the nature of cognition. This paper shows that there is a logical problem with standard functionalist arguments for multiple realizability. These arguments rely essentially on Turing's powerful insights regarding computation. In addressing a possible reply to this criticism, it is further argued that functionalism is not a useful approach for understanding what it is to have a mind. In particular, it is shown that the difficulties involved in distinguishing implementation from function make multiple realizability claims untestable and uninformative. As a result, it is concluded that the role of Turing machines in philosophy of mind needs to be reconsidered.