When the program is run a "workspace" opens (see figure 1), which is a collection of components which can be coupled together in various ways. This is visible as a "desktop," and each component exists as a window in this desktop. Menus in the upper left corner of the workspace allow users to add new components to the desktop and to open and save collections of components. Currently there are three kinds of workspace component: networks, virtual worlds, and gauges, which correspond to neural networks, simulated environments which those network can be attached to, and devices which allow for the visualization of structures in a high dimensional space. Users can add arbitrarily many components to Simbrain. Simbrain opens with a default workspace, and a set of built in workspaces are also provided. To explore these, select "open workspace" from the File menu, and peruse the directories. Collections of components can be saved and reopened.

Figure 1: The Simbrain Workspace

The architecture supporting workspace interactions is being redesigned for the 3.0 release, in order to support more complex environments and component interactions (including distributed environments involving multiple instances of Simbrain communicating over the internet). In the remainder of this section, the three currently available types of workspace component are described in more detail.

Network components, which display neural networks, are the primary Simbrain component (see figure 2). The main design goals of the network component, as noted above, were to make use of familiar interface elements, to allow users to quickly build networks with arbitrary topologies (i.e., networks merging a wide variety of neuron types, including spiking neurons, as well as different network types), and to allow the code to be easily extended. We will see that the generality built in to the network component required that a number of special features be added, including the concept of a "sub-network," and, for future releases, the concept of a logical group of neurons and / or synapses. We will also see that many concepts associated with building networks are such that they can be extended in a modular fashion: neurons, synapses, sub-networks, methods for creating synapses, and methods for laying out groups neurons are among the functions that can currently be customized by the user.

Simbrain worlds are simulated environments, which produce inputs to and accept outputs from nodes in one or more neural networks. The use of the term "world" is a reference to German philosophers who emphasize the role of the "surrounding world" or "lifeworld" in cognition, as well as a growing recognition in cognitive science and philosophy that the study of cognition must incorporate analysis of the bodies and environment within which brains are situated (see Clark and Chalmers 1998 ). As noted above, an early design goal of Simbrain was to be able to study the way environments interact with neural networks in a flexible way: to be able to study neural networks insofar as they are embedded in agents which are embedded in complex, realistic, worlds.

It turns out that the underlying software design issues involved in creating a general framework for linking arbitrarily many neural networks with arbitrarily many environments is non-trivial. It is therefore being completely redesigned for 3.0. However, the 2.0 release does contain two types of world and allows for them to be connected together. The first ("Data world") is a simple table, which facilitates standard uses of environments for neural networks, where a series of input vectors are exposed to a network and corresponding outputs are recorded. The second, "Odor World", is more of a true virtual world; we focus on it here. More sophisticated worlds are being planned for the 3.0 release, and the way in which new worlds are developed and added to Simbrain is being simplified.

An OdorWorld window with four objects and one agent is shown in figure 6.

Figure 6: A Virtual World

The mouse in the center of figure 6 can be attached to a network in Simbrain. Network nodes can be attached to the mouse's sensors and effectors; in this way a neural network can model the response of the mouse to its environment and can control the mouse's movements.

Odor World is inspired by the olfactory system. Objects in the environment emit chemicals which bind to receptors inside a bank of tissue in the nose called the nasal epithelium. Different objects in the environment produce chemicals which have a characteristic impact on the receptors in this tissue ( Laurent 1999 ). Objects can therefore be associated with "smell vectors" which describe the pattern of activity they produce on an agent's nasal epithelium. The chemicals associated with objects diffuse in the atmosphere such that objects typically have a stronger impact on an agent when they are closer to it. This is modeled by scaling the smell vector by distance. When multiple objects are present to an agent overlapping patterns of activation are produced. This is modeled (somewhat inaccurately) by simple vector addition; we take the smell vectors associated with flowers and cheese, in the example above, and add them together. Thus the total pattern of inputs to a creature's olfactory receptors is a sum of scaled smell vectors; what an agent smells depends on what objects are within smelling distance and how far away these objects are. Though the model is based on smell it can be thought of in more general terms as a simple way of modeling distance based interactions with objects. Smell vectors are more generally "stimulus vectors" which describe that object's impact on an arbitrary bank of sensory receptors.

A node in a neural network can be attached to an agent (for example, the mouse in figure 6) by right-clicking on the node and following out a string of submenus which allow the user to choose a world, an agent in that world, and finally a particular "receptor" or "motor command" associated with that agent. When nodes are attached to receptors they become "input nodes" (this is signified by an arrow pointing in to that node). When nodes are attached to motor commands they become "output nodes" (this is signified by an arrow pointing out of that node).

Input nodes are associated with particular receptors on an agent. For example, an input neuron associated with a cheese receptor will be active when the associated agent is near a piece of cheese (in terms of stimulus vectors, receptors are associated with components in a sum of scaled stimulus vectors). How active the input neuron is is based on how far the receptor is from the relevant object. The way the receptors behave can be controlled by double clicking on objects and the agent. By double clicking on an object (e.g. the cheese or the flower in figure 6) its stimulus vector can be edited, the form of distance based scaling can be modified, and noise can be added. For example, we can specify that swiss cheese cause receptor 2 to be extremely active when the agent is nearby, and that its impact will diminish linearly by distance. By double clicking on the agent (the mouse in figure 6) the location of its smell receptors can be edited. This is important when an agent has to be sensitive to the relative location of objects, e.g. if it has to know that an object is to its left rather than its right.

Output nodes are associated with "output commands," which are essentially the motor controls of the agent. When coupling a neuron to an agent, one can choose between two styles of movement: relative movement, and absolute movement. Relative movements are motor commands that tell the agent how to move relative to its current orientation: e.g. move forward or to the right. Absolute movements tell the agent to move in directions that are independent of its orientation, such as move to the north or to the south-west. All movements are scaled based on the activity of the coupled neuron and a fixed movement factor (one for moving straight, one for turning). The larger the activation, the faster the agent. Odor World also provides mechanisms for drawing walls (though this is currently relatively simplistic) and for allowing objects to disappear and reappear.

Using these tools one can connect neural networks with agents in a simple virtual world. To get an intuitive sense of how this works open the simulation sims > two-agents (the default simulation when initially running Simbrain), and press play in one or both of the networks. You can also manually move objects or agents in the world around. Also see the second lesson below.

Plotting and charting capacities are essential to any neural network simulator. As noted above, one specific type of visualization was emphasized in the initial Simbrain design: projection of structures in high dimensional spaces to two dimensions. This is the gauge component, or "high dimensional visualizer." A screenshot of a high dimensional visualizer is shown in figure 7.

Figure 7: A High Dimensional Visualizer

Each blue dot in the visualizer represents a point in an n-dimensional space (i.e., a list of numbers), a vector with n components, which has been projected into two dimensions. The components of these vectors can come from neural activations or synapses in any combinations (so that activation spaces, weight spaces, and combinations of the two can be visualized), and in 3.0 they can arise from any arbitrary workspace parameters. Typically these correspond to all the neural activations of a neural network or one layer of a neural network, so that each blue dot represents a total activation state of a network. The status bar at the bottom shows how many points are currently plotted and what the dimension of the space these points live in is. The blue dots corresponds to states that have occurred previously, and the red dot corresponds to the current state. Lingering over any point presents a tool tip which shows the actual high dimensional vector the point represents. In the example above, the dots represent states of an 8-node neural network, whose current neural activations are, as the tooltip shows, (0, 0, 0, 0, .85, .12, .30, .01, .01). 33 unique states have occurred in this network thus far.

The representation in the visualizer which is attached to the nodes of a neural network present us with a kind of visual history of the states that have occurred in a neural network up to the present point. When running a neural network with a visualizer attached one initially sees these points populate the gauge. Over time, as states that previously occurred recur, a stable pattern develops which shows the set of representations that occur for that network in its current environment. The agent's current state relative to this set of representations is visualized as a red dot that moves within the pattern that has formed.

In the example shown above in figure 7, the gauge represents states of a network in an environment with three objects. 33 network states, each representing a different position of the agent with respect to these objects, have occurred thus far. These representations form a "boquet of arcs," a collection of arcs joined at a single point. Each arc represents one of the objects, the center where the three arcs attach represent absence of any object, and the end-points of the arcs correspond to maximal exposure to one of the objects. As the agent moves towards one of the objects, the red dot moves from the origin out towards one of these ends. Thus, the combination of the gauge and the world allows the user to study the topological and geometric structure of its representation of its environment, a significant capacity for the study of mental representation. To try this simulation open the workspace file sims > bp > 3-2-3_single.sim, create a gauge, and move the agent around the objects.

The gauge allows users to compare the way a high dimensional structure looks under different projections. Three are provided: coordinate, PCA, and Sammon. By comparing the way different projection algorithms present a dataset users can gain qualitative understanding of high-dimensional structures. Consider efforts to project a globe to two dimensions. Even though each projection introduces its own distortions, by comparing them one gets a sense of the three-dimensional structure of our world. Let us consider each of the projection methods currently provided. Coordinate Projection is the simplest of the three, and is best explained by example. If one has a list of datapoints with 40 components each, coordinate projection to two-dimensions simply ignores all but two of these components, which are then used to display the data in two-space. PCA builds on coordinate projection by making use of the "principal axes" of the dataset. The principal axes of an object are the directions in space about which the object is most balanced or evenly spaced. PCA selects the two principal axes along which the dataset is the most spread out and projects the data onto these two axes. The Sammon map is an iterative technique for making interpoint distances in the low-dimensional projection as close as possible to the interpoint distances in the high-dimensional object ( Sammon 1969 ). Two points close together in the high-dimensional space should appear close together in the projection, while two points far apart in the high dimensional space should appear far apart in the projection. When using the Sammon map a play and iterate button are highlighted which allow the user to start and stop the iteration process.

In some cases it is useful to be able to add new points to an existing dataset without running the projection method on the whole dataset again (since this is computation intensive). Methods exist for quickly adding new data points based on data that have already been projected. These methods work best when a certain amount of data has already been collected and projected using, for example, PCA or the Sammon map. These methods can be accessed in the visualizer via the preferences menu.

Beyond high dimensional visualization, there is no charting capacity in Simbrain 2.0, though as part of the 3.0 an open source package will be used to add more standard forms of charting.

Lab: Propagation of Activity

For this example we will not assume any knowledge of the mathematics of neural networks. The goal is simply to get a sense of the way neural networks channel information, the impacts of changing inputs and weights on information flow in a neural network, and at a very broad level, the structure of learning as synaptic modification.

Begin by clearing the workspace, and opening a sample file:

Now try changing the bottom row of the neurons:

Next play with changing weights:

Finally, we modify the synapses so that they can change based on neural activity, which is how learning is modeled in neural networks:

In this example we couple a neural network to a simple environment. The network below controls the mouse on the right. The mouse is similar to a vehicle described by the neuroscientist Valentino Braitenberg, in his book, Vehicles ( Braitenberg 1984 ). The vehicle occurs in chapter 2 of that book, and is described as being "aggressive," because it runs up to the thing it's attracted to as if it's attacking it. In this case the mouse is attracted to the fish.

Again begin by clearing the workspace and opening a sample file:

Play with the mouse and fish:

Get a feel for how the sensor neurons work:

Get a feel for how motor neurons work:

One prevalent way of understanding the behavior of a neural network is in terms of dynamical systems theory. As a first pass, we can think of a dynamical system as a rule which tells us how a system changes its state over time. In the context of neural networks, a dynamical system will tell us how, given an initial assignment of activations to the neurons of a network (and assuming the weights are fixed), those activations will change over time.

To make a neural network which exhibits interesting dynamics, it is useful to create connections which go "backwards," so that activity will flow in loops rather than simply dissipating (as happens in a feed-forward network). This is called recurrence. Here is an example of a recurrent network:

If you put this network in some initial state it may flicker around for a while before "settling down." And it may never settle down at all. It may end up cycling through a few different states over time. Dynamical systems theory gives us a vocabulary for describing these different behaviors.

To clarify these ideas, let us consider some (informally defined) concepts from dynamical systems theory. We begin with the concept of an initial condition, which is just a point in state space which we assume our system begins in.

Initial Condition: the state a system begins in.

For neural networks, this is a specification of values for its nodes. When we set the neurons at some level of activation we are setting the network in an initial condition. Often we arbitrarily choose the initial condition. In Simbrain we do this by selecting all the nodes of a network and then pressing the random button. If we repeatedly press the random button we end up putting the network in a whole bunch of initial conditions.

Dynamical system: A rule which tells us what state a system will be in at any future time, given any initial condition.

A dynamical system can be thought of visually as a recipe for saying, given any point in state space, what points the system will go to at all future times given that it started there: if the system begins here at this point at noon on Thursday, it will be at this other point at 5pm on Friday. And we can do this no matter what initial condition we find our system in. Thus dynamical systems are deterministic, they allow us to completely predict the future based on the present.

The pendulum provides a classical example of a dynamical system: if you start the pendulum off in some beginning state, we can say exactly how it will swing forever in to the future, at least in principle. Most neural networks are also dynamical systems. If you start the neural network off in some state--if you specify values for all its nodes--then based on its update rules and the way it is wired together we can say for all future times how it will behave. Thus, "running" a neural network, in Simbrain by pressing the step or the play button, corresponds to applying the dynamical rule that corresponds to it. A neural network which is predictable in this way is a dynamical system. Can you think of ways to make a neural network not be a dynamical system? We will discuss this below.

Orbit: the set of states that are visited by a dynamical system relative to an initial condition.

So, we put the neural network in some initial state, and then we run the network. It changes state as a result. Each time the neural network changes state we have a new point in activation space. If we look at all the states that result from an initial condition we get an orbit. An orbit describes one of the possible behaviors of a dynamical system over time. You put it in some starting state, and then, since it is a deterministic system, its future is set. The initial condition together with its future states is an orbit.

Invariant set: a collection of orbits.

An invariant set is just a collection of orbits, and it is "invariant" in that if a system begins somewhere in an invariant set, it will stay in that set for all time. The concept of an invariant set allows us to introduce some new concepts which are useful for understanding the different behaviors of a dynamical system.

There are at least two different ways we can classify invariant sets. First, we can classify them by their "topology" or shape. There are two especially interesting shapes for invariant sets:

Fixed point: a state that goes to itself under a dynamical system. The system "stays" in this state forever.

Periodic Orbit: a set of points that the dynamical system visits repeatedly and in the same order. The system cycles through these states over and over. The number of states in a periodic orbit is its "period." These are also known as limit cycles (when they are continuous curves).

Fixed points and periodic orbits are two particular shapes that invariant sets have. Fixed points are single points. Periodic orbits are repeating loops. Note that there are other shapes for invariant sets. For example, some really interesting invariant sets are "chaotic" or "quasi-periodic," but we will not deal with these here.

We can count how many points there are in a periodic orbit, and that corresponds to the "period" of the periodic orbit. For example, a period-2 periodic orbit consists of two points that a neural network cycles through over and over again. A period 3 periodic orbit consists of three points it cycles through, etc. Don't confuse the period of a periodic orbit with the number of periodic orbits a system has. For example, a system might have four periodic orbits, one of period 2, and three of period 3.

Another way we can classify invariant sets is according to the way states nearby the invariant set behave. Sometimes states near an invariant set will tend to go to that set. The invariant set "pulls in" nearby points. These are the stable states, the ones we are likely to see. Other times states near an invariant set will tend to go away from it. These are the unstable states, the ones we are unlikely to see.

Attractor: a state or set of states A with the property that if you are in a nearby state the system will always go towards A.

Fixed points and periodic orbits can both be attractors . These are the states of a system you will tend to observe. These are also known as equilibria.

Repellor: a state or set of states R with the property that if you are in a nearby state the system will always go away from R.

Fixed points and periodic orbits can both be repellors. You are not likely to see these in practice.

One way to think about attractors and repellors is in terms of the possible states of a penny. When the penny is lying on one face or the other, it is in an attractor, a stable fixed point: if you perturb it a little it just falls back to the same place. If you balance the penny on its edge, however, it is on a repelling fixed point. If you perturb it a little in these states it will not go back to the edge-state, but will fall over.

In the chart below, the columns correspond to different topologies or shapes that a invariant set can have. The rows corresponds to the behavior of nearby states.

The items in bold are the ones we have discussed thus far.

Activity

Make a network with four nodes, fully interconnected. To do this:

Now look for attractors and limit cycles. You do this by

Be sure to check for the zero vector! You do this by selecting all neurons and pressing "C" or the clear button.

Throughout this process it can be helpful to have a gauge open. To reset the gauge select all four neurons, right click on one of the neurons, and select set Gauge and the gauge you opened.

Perceptual reversals are a form of visual perception in which, though the total input to the retina remains constant, the perception of that input varies over time. Such phenomena have been recognized for hundreds of years, but have been of particular interest to scientists recently, as a way of studying the neural correlates of consciousness. The reason is that one can use these reversals to distinguish the neurons which code for the stimulus from those which code for the changing percept.

We will look at a neural network model of this process which employs two principles we have seen in other lessons: a winner-take-all competitive structure, and adaptation. Although the model is not directly based on biological data, it does use a neurally inspired architecture. The model is from Hugh Wilson (1999) .

We will consider two special cases of perceptual reversal: ambiguous figures and binocular rivalry.

Ambiguous figures are stimuli which the visual system can interpret in more than one way. Prolonged exposure to such a stimulus can result in an oscillation between these competing interpretations. The most famous example of an ambiguous figure is the Necker cube:

To experience the perceptual reversal fixate on this stimulus. Assuming you can see the two percepts at all, they will begin to oscillate. You will see one, then the other percept over time. Try to estimate the frequency of oscillation. Does it change roughly every 1/2 second, 1 second, 2 seconds?

Another form of perceptual reversal is observed in binocular rivalry. Whereas ambiguous figures correspond to the same image being presented to both eyes, with binocular rivalry a different image is presented to each eye. The conflict here is between these two retinal images. The visual system tries to combine them but they typically conflict in some way, resulting in a perceptual oscillation. Here is a sample stimulus:

Blake and Logothetis (2002) : Visual Competition. Licensed Under OUP Licence

To get the effect you will need to create a septum (a separation between your eyes), so that each image goes to one eye only. You can do this by printing the image, placing an envelope or piece of cardboard between the images, and placing your nose directly over the envelope. You have to sort of unfocus to get the two images to line up. When they do, you should observe an oscillation between the images, as well as moments when the two images merge into a kind of patchwork. Again try to estimate the frequency of oscillation.

The neural network model of perceptual reversal is contained in networks > firingRate > perceptualReversal.net. Feel free to open it now. It is based on a model by Hugh Wilson (1999, pp. 131-134) . The model has the architecture below:

The input neurons, which are clamped, correspond to a stimulus. The output neurons correspond to two possible perceptual interpretations. Note that they are connected by inhibitory neurons and so will end up competing. The basic idea is that we are modeling input channels which feed to competing neural structures. Don't let the identical rates of firing of the inputs fool you. In the case of binocular rivalry, we can just assume that the two inputs code for different patterns, (e.g. left bars and upward bars) both of which are strongly activated.

These are Naka Rushton firing rate models with spike frequency adaptation, so the numbers are supposed to correspond to number of spikes per second, or more generally, level of activation in a neural population. These neuron types are based on integrated differential equations, so you will see an explicit time in the time label.

To see the simulation in action, press play. You will note that the two output neurons, which correspond to the two interpretations, slowly oscillate from one interpretation to the other over time. The goal is to produce an oscillation at the output nodes which matches the frequency of a real perceptual oscillation. To time the simulation, reset the time to 0 by double clicking on the time label. Do this when one output neuron is maximal. Now run the simulation and stop it when the other output neuron is maximal. This gives the time between perceptual interpretations. Compare this with the time elapsed between the reversals when you observed the two inputs.

The mechanism works as follows. The two output neurons produce a winner take all structure, since they mutually inhibit one another. But they also have spike frequency adaptation. When the simulation begins, notice that one neuron already has an activation of 1, while the other's activation is 0. This slight bias allows that neuron to initially win the competition. However, as that neuron continues to fire, adaptation kicks in, and it reduces its activity, which in turn allows the other neuron to win the competition. But then its adaptation begins to kick in, and the reverse process occurs.

To change the frequency of perceptual reversal change the firing rate time constants at the output nodes (select them both and double click to change their parameters), and modify the time step accordingly (as you make the time constants smaller, and thereby increase the time it takes for perceptual reversals to occur, you will also have to lower the time step of integration to avoid numerical errors).

To visualize these dynamics try opening a gauge and setting it to gauge the two output nodes. I recommend you set the point size (in preferences > graphics preferences) to .8, and "only add new points if this far from any other point" to .8 (in preferences > general preferences) in the gauge.

You will observe a limit cycle with a figure-8 shape, which looks like this:

The two ends correspond to the two perceptual interpretations. Notice that the system spends a longer time on the edges of the cycle, which correspond to the different interpretations, than in the transitional portion of the figure in the middle.