Edited by: Runchun Mark Wang, Western Sydney University, Australia
Reviewed by: Hesham Mostafa, Intel, United States; André van Schaik, Western Sydney University, Australia
This article was submitted to Neuromorphic Engineering, a section of the journal Frontiers in Neuroscience
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In neuromorphic engineering, neural populations are generally modeled in a bottom-up manner, where individual neuron models are connected through synapses to form large-scale spiking networks. Alternatively, a top-down approach treats the process of spike generation and neural representation of excitation in the context of minimizing some measure of network energy. However, these approaches usually define the energy functional in terms of some statistical measure of spiking activity (ex. firing rates), which does not allow independent control and optimization of neurodynamical parameters. In this paper, we introduce a new spiking neuron and population model where the dynamical and spiking responses of neurons can be derived directly from a network objective or energy functional of continuous-valued neural variables like the membrane potential. The key advantage of the model is that it allows for independent control over three neuro-dynamical properties: (a) control over the steady-state population dynamics that encodes the minimum of an exact network energy functional; (b) control over the shape of the action potentials generated by individual neurons in the network without affecting the network minimum; and (c) control over spiking statistics and transient population dynamics without affecting the network minimum or the shape of action potentials. At the core of the proposed model are different variants of Growth Transform dynamical systems that produce stable and interpretable population dynamics, irrespective of the network size and the type of neuronal connectivity (inhibitory or excitatory). In this paper, we present several examples where the proposed model has been configured to produce different types of single-neuron dynamics as well as population dynamics. In one such example, the network is shown to adapt such that it encodes the steady-state solution using a reduced number of spikes upon convergence to the optimal solution. In this paper, we use this network to construct a spiking associative memory that uses fewer spikes compared to conventional architectures, while maintaining high recall accuracy at high memory loads.
Spiking neural networks that emulate neural ensembles have been studied extensively within the context of dynamical systems (Izhikevich,
An alternative to this bottom-up approach is a top-down approach that treats the process of spike generation and neural representation of excitation in the context of minimizing some measure of network energy. The rationale for this approach is that physical processes occurring in nature have a tendency to self-optimize toward a minimum-energy state. This principle has been used to design neuromorphic systems where the state of a neuron in the network is assumed to be either binary in nature (spiking or not spiking) (Jonke et al.,
In Gangopadhyay and Chakrabartty (
We first remap the synaptic interactions in a standard spiking neural network in a manner that the solution (steady-state attractor) could be encoded as a first-order condition of an optimization problem. We show that this network objective function or energy functional can be interpreted as the total extrinsic power required by the remapped network to operate, and hence a metric to be minimized.
We then introduce a dynamical system model based on Growth Transforms that evolves the network toward this steady-state attractor under the specified constraints. The use of Growth Transforms ensures that the neuronal states (membrane potentials) involved in the optimization are always bounded and that each step in the evolution is guaranteed to reduce the network energy.
We then show how gradient discontinuity in the network energy functional can be used to modulate the shape of the action potential while maintaining the local convexity and the location of the steady-state attractor.
Finally, we use the properties of Growth Transforms to generalize the model to a continuous-time dynamical system. The formulation will then allow for modulating the spiking and the population dynamics across the network without affecting network convergence toward the steady-state attractor.
We show that the proposed framework can be used to implement a network of coupled neurons that can exhibit memory, global adaptation, and other interesting population dynamics under different initial conditions and based on different network states. We also illustrate how decoupling transient spiking dynamics from the network solution and spike-shapes could be beneficial by using the model to design a spiking associative memory network that can recall a large number of patterns with high accuracy while using fewer spikes than traditional associative memory networks. This paper is also accompanied by a publicly available software implementing the proposed model (Mehta et al.,
In this section, we present the network energy functional by remapping the synaptic interactions of a standard spiking neural network and then propose a Growth Transform based dynamical system for minimizing this objective. For the rest of the paper, we will follow the mathematical notations as summarized below.
Real scalar variable | |
Real-valued vector with |
|
Real-valued matrix with |
|
a sequence of scalar variables |
|
𝔼 |
Empirical expectation of a sequence |
i.e., |
|
Empirical expectation estimated over an asymptotically infinite window, | |
i.e., |
|
Real-valued vector with |
|
𝔼 |
Empirical expectation of |
ℝ |
Vector space spanned by |
| |
Absolute value of a scalar |
|| |
|
Transpose of the vector |
|
Inner product between the vectors |
|
Gradient vector |
In generalized threshold models like the Spike Response Model (Gerstner and Kistler,
where
We further enforce that the membrane potentials are bounded by
Note that in biological neural networks, the membrane potentials are also bounded (Wright,
If Ψ(.) was a smooth function of the membrane potential,
where
where
where
subject to the bound constraint |
The network energy functional
In order to solve the energy minimization problem given in (8) under the constraints given in (2), we first propose a dynamical system based on polynomial Growth Transforms. We also show how the dynamical system evolves over time to satisfy (7) as a first-order condition.
Growth Transforms are multiplicative updates derived from the well-known Baum-Eagon inequality (Baum and Sell,
Discrete-time Growth Transform dynamical system (Proof in
satisfies the following criteria for all time-indices |
Considering the
where
Then asymptotically from (1), and as shown in
where
Therefore, (15) can be written as
Thus as long as
The dynamical system presented in (9) ensures that the steady-state neural responses
When there is no external stimulus
where the trans-impedance parameter
In order to show the effect of Ψ(.) on the nature of the solution, we plot in
In
This implies that asymptotically the network exhibits limit-cycles about a single attractor or a fixed-point such that the time-expectations of its state variables encode this optimal solution. A similar stochastic first-order framework was used in Gore and Chakrabartty (
where
The penalty function
Discrete-time GT spiking neuron model.
For a network of M neurons with state variables |
λ is a fixed current parameter such that |
The composite spike response of the |
As explained previously, the penalty term
Thus the average spiking activity of the
we have
Multiplying (25) on both sides by
where we have used the relation (19). Equation (27) indicates that through a suitable choice of the trans-impedance parameter
where the average spiking activity tracks the stimulus. Thus, by defining the coupling matrix in various ways, we can obtain different encoding schemes for the network.
The remapping from standard coupled conditions of a spiking neural network to the proposed formulation admits a geometric interpretation of neural dynamics. Similar to the network coding framework presented in Gangopadhyay and Chakrabartty (
Like a Hessian, if we assume that the matrix
where
Single neurons show a vast repertoire of response characteristics and dynamical properties that lend richness to their computational properties at the network level. Izhikevich (
Complete continuous-time GT spiking neural network (Proof in Appendix D).
For a network of M neurons with state variables |
where
|
λ is a fixed current parameter such that |
|
The composite spike response of the |
where the trans-impedance parameter |
The operation of the proposed neuron model is therefore governed by two sets of dynamics: (a) minimization of the network energy functional
Decoupling of network solution, spike shape and response trajectory using the proposed model. Different modulation functions lead to different steady-state spiking dynamics under the same energy contour.
The proposed approach enables us to decouple the three following aspects of the spiking neural network:
Fixed points of the network energy functional, which depend on the network configuration and external inputs;
Nature and shape of neural responses, without affecting the network minimum; and
Spiking statistics and transient neural dynamics at the cellular level, without affecting the network minimum or spike shapes.
This makes it possible to independently control and optimize each of these neuro-dynamical properties without affecting the others. The first two aspects arise directly from an appropriate selection of the energy functional and were demonstrated in section 2.2.1. In this section, we show how the modulation function, in essence, loosely models cell excitability, and can be varied to tune transient firing statistics based on local and/or global variables. This allows us to encode the same optimal solution using widely different firing patterns across the network, and have unique potential benefits for neuromorphic applications. Codes for the representative examples given in this section are available at Mehta et al. (
We first show how we can reproduce a number of single-neuron response characteristics by changing the modulation function τ
We will subsequently extend these dynamics to build coupled networks with interesting properties like memory and global adaptation for energy-efficient neural representation. The results reported here are representative of the types of dynamical properties the proposed model can exhibit, but are by no means exhaustive. Readers are encouraged to experiment with different inputs and network parameters in the software (MATLAB©) implementation of the Growth Transform neuron model (Mehta et al.,
When stimulated with a constant current stimulus
Bursting neurons fire discrete groups of spikes interspersed with periods of silence in response to a constant stimulus (McCormick et al.,
where τ1 > τ2 > 0 s,
When presented with a prolonged stimulus of constant amplitude, many cortical cells initially respond with a high-frequency spiking that decays to a lower steady-state frequency (Connors and Gutnick,
where
is a compressive function that ensures 0 ≤ τ
When the baseline input is set slightly negative so that the fixed point is below the threshold, the neuron works like a leaky integrator as shown in
We can extend the proposed framework to a network model where the neurons, apart from external stimuli, receive inputs from other neurons in the network. We begin by considering
with the compressive-function ϕ(.) given by (37). Equation (38) ensures that
Apart from the pre-synaptic adaptation that changes individual firing rates based on the input spikes received by each neuron, neurons in the coupled network can be made to adapt according to the global dynamics by changing the modulation function as follows
with the compressive-function ϕ(.) given by (37). The new function
where
In order to outline the premises of the next few experiments on population dynamics using the geometric interpretation outlined in section 2.3, we consider a small network of neurons on a two-dimensional co-ordinate space, and assign arbitrary inputs to the neurons. A Gaussian kernel is chosen for the coupling matrix
This essentially clusters neurons with stronger couplings between them closer to each other on the co-ordinate space, while placing neurons with weaker couplings far away from each other. A network consisting of 20 neurons is shown in
The Growth Transform neural network inherently shows a number of encoding properties that are commonly observed in biological neural networks (Rieke et al.,
These coding schemes can be interpreted under the umbrella of network coding using the same geometric representation as considered above. Here, the responsiveness of a neuron is closely related to its proximity to the hyperplane. The neurons which exhibit more spiking are located at a greater distance from the hyperplane. We see from
The encoding of a stimulus in the spatiotemporal evolution of activity in a large population of neurons is often represented in neurobiological literature by a unique trajectory in a high-dimensional space, where each dimension accounts for the time-binned spiking activity of a single neuron. Projection of the high-dimensional activity to two or three critical dimensions using dimensionality reduction techniques like Principal Component Analysis (PCA) and Linear Discriminant Analysis (LDA) have been widely used across organisms and brain regions to shed light on how neural population response evolves when a stimulus is delivered (Friedrich and Laurent,
For the same network as above, we start with the simplest possible experiment, starting from the same baseline, and perturbing the stimulus vector in two different directions. This pushes the network hyperplane in two different directions, exciting different subsets of neurons, as illustrated in
As illustrated in
due to the presence of more than one attractor state. We demonstrate this by considering two different stimulus histories in a network of four neurons, where a stimulus “Stim 1a” precedes another stimulus “Stim 2” in
Stimulus response for a 4-neuron network with different stimulus histories for:
Associative memories are neural networks which can store memory patterns in the activity of neurons in a network through a Hebbian modification of their synaptic weights; and recall a stored pattern when stimulated with a partial fragment or a noisy version of the pattern (Cutsuridis et al.,
Our network comprises
where
When the network is made to globally adapt according to the system dynamics, the steady-state trajectories can be encoded using very few spikes.
where
where dist
To estimate the capacity of the network, we calculate the mean recall accuracy over 10 trials for varying number of stored patterns, both with and without global adaptation.
Ensemble plots showing
Note that the recall accuracy using global adaptation deteriorates faster for > 175 patterns. The proposed decoding algorithm, which determines the recall accuracy, takes into account the mean spiking rates, inter-spike intervals and changes in spike rates. It is possible that as the number of spikes is reduced through the use of global adaptation, the information encoded in first-order differences (inter-spike intervals or spike rates) may not be sufficient to encode information at high fidelity, resulting in the degradation in recall accuracy when the number of patterns increased. However, augmenting the decoding features with higher-order differences in inter-spike intervals or spike rates may lead to an improved performance for higher storage.
Aside from pattern completion, associative networks are also commonly used for identifying patterns from their noisy counterparts. We use a similar associative memory network as above to classify images from the MNIST dataset which were corrupted with additive white Gaussian noise at different signal-to-noise ratios (SNRs), and which were, unlike in the previous case, unseen by the network before the recall phase. The network size in this case was
The test accuracies and mean spike counts for a test image are plotted in
This paper introduces the theory behind a new spiking neuron and population model based on the Growth Transform dynamical system. The system minimizes an appropriate energy functional under realistic physical constraints to produce emergent spiking activity in a population of neurons. The proposed work is the first of its kind to treat the spike generation and transmission processes in a spiking network as an energy-minimization problem involving continuous-valued neural state variables like the membrane potential. The neuron model and its response are tightly coupled to the network objective, and are flexible enough to incorporate different neural dynamics that have been observed at the cellular level in electrophysiological recordings.
The paper is accompanied by a software tool (Mehta et al.,
In this regard, machine learning models are primarily developed with the objective of minimizing the error in inference by designing a loss function that captures dependencies among variables, for example, features and class labels. Learning in this case, as pointed out in LeCun et al. (
The network energy functional bears similarity with the Ising Hamiltonians used in Hopfield networks (Hopfield,
The energy-based formulation described in section 2.1 could also admit other novel interpretations. For instance, for the form of Ψ(.) considered in (18), the barrier function can be rewritten as
Also, if we consider the spike response as a displacement current, we can write
where
Thus according to this interpretation, the communication between neurons takes place using current waveforms, similar to integrate-and-fire models, and the current waveforms can be integrated at the post-synaptic neuron to recover the membrane potential. Note that the remapping between
In the proposed neuron model, we abstracted out the essential aspects of spike generation and transmission that can replicate neural dynamics, and remapped synaptic interactions to an energy-based framework. As a result of the remapping procedure, the coupling matrix
A key advantage of the proposed framework is that it enables the decoupling of the three neurodynamical parameters - network solution, spike shapes and transient dynamics. Thus while the solution to the energy functional is determined by the coupling matrix
A hybrid spiking network comprising neurons of different types (tonic spiking, bursting, non-spiking, etc.), as illustrated in section 3.1. The network would still converge to the same solution, but the spiking dynamics across the network could be exploited to influence factors such as speed, energy efficiency and noise-sensitivity of information processing.
Optimization of some auxiliary network parameter, e.g., the total spiking activity. A related example (although not optimized w.r.t. any objective function) was illustrated in section 3.6 for a simple associative network. In this example, the network recalled the same set of patterns and classified MNIST images using two different time-evolutions of the modulation function corresponding to the presence and absence of global adaptation. In this case, it had the benefit of using fewer spikes to achieve better recall when a modified decoding metric was used.
Modeling the effect of neurotransmitters and metabolic factors that have been known to affect the properties, activity and functional connectivity of populations of neurons. These factors endow the same network with the flexibility to generate different output patterns and produce different behaviors for the same stimulus (Hasselmo,
Modeling the effect of diffusion processes or glial processes, that have been known to modulate response properties and synaptic transmission in neurons, influencing information processing and learning in the brain (Clarke and Barres,
All datasets generated for this study are included in the article/
AG and SC contributed to the conception and design of the study and wrote the first draft of the manuscript. AG and DM conducted the simulations. AG, DM, and SC designed the MATLAB interface. All authors contributed to the manuscript revision, read and approved the submitted version.
The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.
The authors would like to thank Dr. Kenji Aono at the Electrical and Systems Engineering department, Washington University, for developing a GPU version of the GT neural network model which is also included with the accompanying software toolbox (Mehta et al.,
This manuscript has been released as a pre-print at Gangopadhyay et al. (
The Supplementary Material for this article can be found online at: