![Graded Patterns in Attractor Networks
Tristan Webb Supervisor: Jianfeng Feng Co-Supervisor: Edmund Rolls
Complexity Science DTC, Computational Biology Research Group
University of Warwick
Summary
We demonstrate how noise can exist in a neural network as large as the brain. Graded firing patterns allow us to tune noise levels in the
engineering of neural networks. The levels of noise in the brain may change with age and play a functional role in the retrieval of memory.
Attractor Neural Networks Graded Patterns
Neural coding, and its relationship to behavior, is heavily researched The network was simulated numerically for a time period of four sec-
in many areas of neuroscience. Attractor networks are a demonstra- onds. We present the network two different periods of different exter-
tion of how decisions, memories, and other cognitive representations nal stimulus levels: first a base period, and later a cue period. During
can be encoded in a firing pattern (or set of active neurons) in a neu- the cue period the qualitative firing pattern in the network is sporadic
ral network. and uneven. When cues are applied, the firing rate for the neurons in
An attractor network receives sensory information to the network a winning decision pool is raised to through positive feedback, while
through connections known as synapses. The network is character- the other pool is suppressed through increased inhibition.
ized by recurrent collateral synapses providing feedback to neurons. Uniform Graded
Recurrent synaptic activity will cause the firing patterns in the network Final Second Mean Neuron Rates Final Second Mean Neuron Rates
60 60
to persist even after the input is removed. Winning Pool Winning Pool
Learning occurs through the modification of synaptic strengths (wij , 50 Losing Pool 50 Losing Pool
where i is the ith neuron and j is the jth synapse). An associative 40 40
Firing Rate (Hz)
Firing Rate (Hz)
learning (Hebbian) rule can create the correct structure for the re- 30 30
call of information. This type of learning strengthens connections 20 20
between neurons that are simultaneously active.
10 10
The network dynamics can be thought of as a gradient descent to-
wards a local minimum in an energy landscape. When the network 00 5 10 15 20 25 30 35 40 00 5 10 15 20 25 30 35 40
Neuron Number Neuron Number
has reached this minimum the learned pattern is recalled. The en-
ergy is defined as We imposed uniform and graded firing patterns on the network by
1 External Inputs selecting the distribution of the recurrent weight for each of the deci-
E =− (yi − < y >)(yj − < y >) sion pools. To achieve a uniform firing pattern, weights were all set
2
ij Recurrent firing
yj
Dendrites to the same value w+ = 2.1. Graded firing patterns were achieved
Recurrent
where yi is the firing of the ith neu- wij collateral
by conforming weights to a discrete exponential-like distribution with
ron, < y > is the population’s mean fir- synapses mean value w+ ≈ 2.1.
ing rate. Fixed points in attractor net- Cell bodies
works can correspond to a spontaneous Output firing
Results
state (where all neurons have a low fir- yi
Graded simulations were more likely to jump to a decision
ing rate), or a persistent state in which a early. This could be caused by decreased stability of the
subset of neurons have a high firing rate. spontaneous state. Changes in reaction time distributions
are statistically significant and the decrease in reaction time
Network Dynamics is robust across different firing rates of the winning pool.
Variability in the system increases when
Neurons in simulations use Integrate-and-Fire (IF) dynamics to de- Reaction Times vs Firing Rates
1100 graded patterns are introduced. Here
scribe the membrane potential of neurons. We chose biologically
1000 we use the Fano factor to compute trial
realistic constants to obtain firing rates that are comparable to ex-
Reaction Time (msec)
900 to trial variability of membrane potentials
perimental measurements of neural activity. IF neurons integrate 800 across simulations. The Fano factor is
synaptic current into a membrane potential, and then fire when the 700 calculated from the variance in the poten-
membrane potential reaches a voltage threshold. 600 Graded Simulations
Uniform Simulations
tial measured in a window with temporal
The synaptic current flowing into each neuron is described in terms 500
26 27 28 29 30 31 32 33 34
Winning Pool Final Second Firing Rate (Hz) length T and expressed as a function of
of neurotransmitter components. The four families of receptors used
time,
are GABA, NDMA, AMPArec , and AMPAext . The neurotransmitter re- Average Fano Factor of Membrane Potential
0.005 Tr
leased from a presynaptic excitatory neuron are AMPArec and NMDA, [Vi,n (T ) − Vi (T ) ]2
while inhibitory neurons transmit GABA currents. Each neuron re- 0.004
F (T ) = n ,
ceives external input through a spike train modeled by a Poisson pro- 0.003
Fano Factor
Vi (T )
cess with rate λi = 3.0Hz. 0.002
where Vi (T ) is the average potential of
Synaptic current flowing into a neuron is given by the following equa- 0.001 Graded Simulations neuron i in the time window, and Tr is the
tion, where each term on the RHS refers to the current from one class Uniform Simulations
0.000 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 number of trials.
of neurotransmitter, Time (seconds)
Isyn (t) = IGABA(t) + INDMA(t) + IAMPA,rec (t) + IAMPA,ext (t) Conclusion
The transition time to an attractor state, or reaction time, is
Architecture decreased when neurons fire in a more biologically realistic
We structure the network by establishing the strength of interactions pattern.
between two decision pools, D1 & D2, to be values that could occur There is greater variability in the system’s states over time when
through associative learning. graded patterns are introduced.
We state that increased variance in synaptic input to each neuron can
Non-Specific 1 be thought of as increased noise in the system. Conceptually, graded
Inhibitory Excitatory
Neurons Neurons
Blowup showing sub-populations
of exictatory neurons
patterns are more noisy because recurrent synaptic input to neurons
w+
D1
w−
D2
w+
will vary across the population.
As neural networks become larger, noise will invariably become
lower. However, when we consider the situation in brain, even though
Neurons in the same decision pool are connected to each other with the network is large, there is still significant noise in the system. We
an strong average weight w+, and are connected to the other excita- present the hypothesis that this noise is due in part to graded firing
tory pools with an weak average weight w−. pattens. Further work will explore this analytically.
Complexity DTC - University of Warwick - Coventry, UK Mail: tristan.webb@warwick.ac.uk WWW: http://warwick.ac.uk/go/tristanwebb](https://image.slidesharecdn.com/dtcposter-12771179465742-phpapp02/85/Graded-Patterns-in-Attractor-Networks-1-320.jpg)
Graded Patterns in Attractor Networks
- 1. Graded Patterns in Attractor Networks Tristan Webb Supervisor: Jianfeng Feng Co-Supervisor: Edmund Rolls Complexity Science DTC, Computational Biology Research Group University of Warwick Summary We demonstrate how noise can exist in a neural network as large as the brain. Graded firing patterns allow us to tune noise levels in the engineering of neural networks. The levels of noise in the brain may change with age and play a functional role in the retrieval of memory. Attractor Neural Networks Graded Patterns Neural coding, and its relationship to behavior, is heavily researched The network was simulated numerically for a time period of four sec- in many areas of neuroscience. Attractor networks are a demonstra- onds. We present the network two different periods of different exter- tion of how decisions, memories, and other cognitive representations nal stimulus levels: first a base period, and later a cue period. During can be encoded in a firing pattern (or set of active neurons) in a neu- the cue period the qualitative firing pattern in the network is sporadic ral network. and uneven. When cues are applied, the firing rate for the neurons in An attractor network receives sensory information to the network a winning decision pool is raised to through positive feedback, while through connections known as synapses. The network is character- the other pool is suppressed through increased inhibition. ized by recurrent collateral synapses providing feedback to neurons. Uniform Graded Recurrent synaptic activity will cause the firing patterns in the network Final Second Mean Neuron Rates Final Second Mean Neuron Rates 60 60 to persist even after the input is removed. Winning Pool Winning Pool Learning occurs through the modification of synaptic strengths (wij , 50 Losing Pool 50 Losing Pool where i is the ith neuron and j is the jth synapse). An associative 40 40 Firing Rate (Hz) Firing Rate (Hz) learning (Hebbian) rule can create the correct structure for the re- 30 30 call of information. This type of learning strengthens connections 20 20 between neurons that are simultaneously active. 10 10 The network dynamics can be thought of as a gradient descent to- wards a local minimum in an energy landscape. When the network 00 5 10 15 20 25 30 35 40 00 5 10 15 20 25 30 35 40 Neuron Number Neuron Number has reached this minimum the learned pattern is recalled. The en- ergy is defined as We imposed uniform and graded firing patterns on the network by 1 External Inputs selecting the distribution of the recurrent weight for each of the deci- E =− (yi − < y >)(yj − < y >) sion pools. To achieve a uniform firing pattern, weights were all set 2 ij Recurrent firing yj Dendrites to the same value w+ = 2.1. Graded firing patterns were achieved Recurrent where yi is the firing of the ith neu- wij collateral by conforming weights to a discrete exponential-like distribution with ron, < y > is the population’s mean fir- synapses mean value w+ ≈ 2.1. ing rate. Fixed points in attractor net- Cell bodies works can correspond to a spontaneous Output firing Results state (where all neurons have a low fir- yi Graded simulations were more likely to jump to a decision ing rate), or a persistent state in which a early. This could be caused by decreased stability of the subset of neurons have a high firing rate. spontaneous state. Changes in reaction time distributions are statistically significant and the decrease in reaction time Network Dynamics is robust across different firing rates of the winning pool. Variability in the system increases when Neurons in simulations use Integrate-and-Fire (IF) dynamics to de- Reaction Times vs Firing Rates 1100 graded patterns are introduced. Here scribe the membrane potential of neurons. We chose biologically 1000 we use the Fano factor to compute trial realistic constants to obtain firing rates that are comparable to ex- Reaction Time (msec) 900 to trial variability of membrane potentials perimental measurements of neural activity. IF neurons integrate 800 across simulations. The Fano factor is synaptic current into a membrane potential, and then fire when the 700 calculated from the variance in the poten- membrane potential reaches a voltage threshold. 600 Graded Simulations Uniform Simulations tial measured in a window with temporal The synaptic current flowing into each neuron is described in terms 500 26 27 28 29 30 31 32 33 34 Winning Pool Final Second Firing Rate (Hz) length T and expressed as a function of of neurotransmitter components. The four families of receptors used time, are GABA, NDMA, AMPArec , and AMPAext . The neurotransmitter re- Average Fano Factor of Membrane Potential 0.005 Tr leased from a presynaptic excitatory neuron are AMPArec and NMDA, [Vi,n (T ) − Vi (T ) ]2 while inhibitory neurons transmit GABA currents. Each neuron re- 0.004 F (T ) = n , ceives external input through a spike train modeled by a Poisson pro- 0.003 Fano Factor Vi (T ) cess with rate λi = 3.0Hz. 0.002 where Vi (T ) is the average potential of Synaptic current flowing into a neuron is given by the following equa- 0.001 Graded Simulations neuron i in the time window, and Tr is the tion, where each term on the RHS refers to the current from one class Uniform Simulations 0.000 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 number of trials. of neurotransmitter, Time (seconds) Isyn (t) = IGABA(t) + INDMA(t) + IAMPA,rec (t) + IAMPA,ext (t) Conclusion The transition time to an attractor state, or reaction time, is Architecture decreased when neurons fire in a more biologically realistic We structure the network by establishing the strength of interactions pattern. between two decision pools, D1 & D2, to be values that could occur There is greater variability in the system’s states over time when through associative learning. graded patterns are introduced. We state that increased variance in synaptic input to each neuron can Non-Specific 1 be thought of as increased noise in the system. Conceptually, graded Inhibitory Excitatory Neurons Neurons Blowup showing sub-populations of exictatory neurons patterns are more noisy because recurrent synaptic input to neurons w+ D1 w− D2 w+ will vary across the population. As neural networks become larger, noise will invariably become lower. However, when we consider the situation in brain, even though Neurons in the same decision pool are connected to each other with the network is large, there is still significant noise in the system. We an strong average weight w+, and are connected to the other excita- present the hypothesis that this noise is due in part to graded firing tory pools with an weak average weight w−. pattens. Further work will explore this analytically. Complexity DTC - University of Warwick - Coventry, UK Mail: tristan.webb@warwick.ac.uk WWW: http://warwick.ac.uk/go/tristanwebb